Divisibility of Magnitude in De Generatione et Corruptione I.2

GC I.2 is principally dedicated to the presentation and subsequent refutation of an argument alleging that magnitude is indivisible. Offered at 316a14-316b18, the argument has the form of a reductio ad absurdum and appears to prove that the divisibility thesis implies absurd consequences. The reductio is refuted later in the chapter (317a2-12), with the refutation becoming a vehicle for a proper characterization of what it is for magnitude to be divisible, or so I ague. ‘Magnitude’ here refers both to bodily and mathematical magnitude, by which it is meant lines, planes and geometrical solids.

Since Philoponus, the reductio is thought to be an argument by Democritus and Aristotle’s aim in this chapter to refute the Democritean ontology.1 Commentators insist on attributing it to Democritus, even though they acknowledge that it relies heavily on Aristotelian doctrine.2 Skeptical voices have been raised, but they have been few and unconvincing,3 the chief reason being that they fail to present Aristotle with an alternative motivation for rehearsing and refuting the argument.

Two aporiai form the backbone of the chapter. The first, at 315b20-3, warns of a threat to the divisibility thesis and outlines the aim of the chapter as being to remove it. The second, at 316b19-27, explains this threat. Correctly understood, the first aporia also lays it down that the atomist ontology is already refuted, with no prospect of being resurrected. It is unlikely to be the threat of which that aporia warns. Still a further refutation of atomism would not therefore serve the chapter’s aims. Moreover, the chapter’s use of the terms ‘σῶµα’ and ‘µέγεθος’ casts further doubt on the claim that the reductio is by Democritus. Fimally, the argument is all too obviously flawed to have been used by Democritus, or anyone for that matter, in support of atomism. To the extent that GC I.2 has been used as evidence for

1See Joachim (1922), 76; Williams (1982); Luria (1933), 129-133; 23 ff.; Schramm (1962), 245-264; Miller (1982), 87-111; Barnes (1982); 356 ff.; Makin (1993), 49-62; Furley (1967), 85-87. Also Sedley (2004): “Why this elaborately contrived refutation of Democritean atomism?” (p. 82). 2 Rosen & Malink (2012) correctly point out that the reductio is based on a view Aristotle says was “discovered by himself, as the result of substantial theoretical work on his part”. But neither can resist the temptation of attributing the argument to Democritus: the argument says Rosen & Malink “can be regarded as a combination of Democritean and Aristotelian elements” (p. 31). They do not explain what the Democritean element is, though one may assume they take it to be the argument’s apparent support for atomism. 3 Cherniss (1935), 113, remarks that the reductio is offered as a reasoning that makes people think there are atomic bodies. Mau (1954), 25-6, is also skeptical that it is by Democritus.

Democritean doctrine and in particular the view that Democritus rejected mathematical divisibility,4 the results of this part of the paper will have a bearing on that question too.

The reductio could be an argument by atomist sympathizers of Speusippean or Xenocratean persuasion and familiar with Aristotelian doctrine. It could even be by Aristotle himself. Be that as it may, I worry that focusing on it as Democritean completely misses the key issue with which Aristotle is here most anxious to grapple. As the second of the two aporiai mentioned above makes clear, this is to remove a threat to the grounds for the position of Physics III.6 that the infinite is an attribute of magnitude (206b5-9). Prima facie, the difficulty is a reductio argument alleging that the proposed grounding is incoherent, thereby threatening the divisibility thesis. But this is a ploy Aristotle uses in order to expose a view on point assumed by the reductio that conflicts with his conception of magnitude and sharply to dissociate it from his own, which he then articulates. On that basis, he is able coherently to ground the position of Physics III.6 that the infinite is an attribute of magnitude.

I In the first chapter of the GC Aristotle outlines important goals for this work. Chief among them is to offer accounts of unqualified generation and corruption (περὶ γενέσεως καὶ φθορᾶς τῆς ἁπλῆς) doing justice to the intuition that they are radically different form each other and, no less importantly, each is likewise different from alteration (ἀλλοίωσιν). Properly being able to account for all three and for bringing out their differences, he considers an adequacy test for any physical theory (314b12-17).5

Generation and corruption are discussed in chapter I.3 and alteration in I.4, which naturally raises this question; why does the discussion of the items on the agenda broadcasted in the first chapter not begin here? What is the important issue with which must GC I.2 must deal before the announced program can be set under way? GC I.2 begins by recalling the extended criticisms GC I.1 levels against previous thinkers for failing to account properly for

4‘Mathematical divisibility’ refers to the view that what Aristotle calls intelligible magnitude or intelligible body, namely the lines, planes and solids studied by geometers are divisible. If Democritus denies that atoms considered as mathematical solids are divisible or that parts can be marked in an atom, he is likely to be committed to mathematical indivisibility. 5 So in GC I.1 Aristotle offers detailed criticisms of previous thinkers for failing to account for generation, corruption and alteration. Some, he says, claim that everything is really one (ἕν τι τὸ πᾶν), and consequently think of generation and corruption as being merely alteration; others, as for instance Anaxagoras (314a13-6) and Empedocles (314b6-8) fail to distinguish between generation, corruption and alteration even though they postulate the existence of more than one elements.

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generation, corruption and alteration. But unlike GC I.1, it singles out one thinker for praise. Democritus is applauded for offering accounts of generation and corruption and appreciating the difference between them as wells as the difference of each from alteration. On Democritus’ account, as Aristotle reports it, generation happens by association (σύγκρισις) and corruption by dissociation (διάκρισις) (315b6-9). There is generation when atoms associate to form stable configurations and corruption when they dissociate, with the configurations they previously formed being destroyed. In alteration, we have merely a change of position of the atoms comprising a particular configuration they jointly form, one that does not, however, bring about the destruction of that configuration.6

Here we have, one might think, the motivation for postponing the account of generation and corruption to the next chapter. Democritus offers competing accounts of generation, corruption and alteration. If atomism is left standing, Aristotle’s forthcoming account will have a competitor. Consequently, one might think, this chapter is dedicated to refuting Democritean and Leucippean atomism7 in order for Aristotle’s account to get the spoils. But this interpretation faces an obstacle. Nowhere in this chapter does Aristotle signal that he needs to refute atomism to serve his purpose. On the contrary, he signals that he has refuted atomism already. He says early on and long before the reductio that the Democritean ontology defies reason (πολλὴν ἔχει ἀλογίαν; 315b33). Also, the first horn of the aporia at 315b20-3, states that if generation happens by association, impossible consequences follow. These impossible consequences are certain to include the atomist ontology. Generation as association is the account of the Atomists, and of no other as far as we know. In context, then, generation as association implies atomism, and so what Aristotle says here, if only by implication, is that he has already shown atomism to be impossible. And he unequivocally confirms that he is already done with proving that when he says a little later in the chapter, at 316b16-8, that he has proved in other works that atomism is impossible.8 GC I.2 presupposes,

6 Democritus would presumably account for natural growth (αὔξησις) and diminution (φθίσις) by means of a similar mechanism he accounts for generation and corruption. A configuration grows bigger when additional atoms join in, or smaller when some of the atoms forming it depart, while the configuration itself persists, becoming larger or smaller respectively. 7 With ‘atomism’ I refer to the views of Democritean and Leucippean and by ‘Atomists’ to Democritus and Leucippus. I assume there is no difference between the views of the two men. So, using the name of only one of them, when I do, is no indication that I’m referring to his view as opposed to those of the other. 8Οὐ µὴν ἀλλὰ καὶ ταῦτα θεµένοις οὐχ ἧττον συµβαίνει ἀδύνατα. Ἔσκεπται δὲ περὶ αὐτῶν ἐν ἑτέροις. See for instance Phys. VI and more importantly De Caelo 303a3 ff. One of the previous work referred to here could well be the De Caelo, of which the Generatione et Corruptione, is presumably the sequel. If so, there is still less of a reason to engage anew in a recently undertaken task.

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and says in so many words in several passages, that atomism is both unworkable and already refuted, which makes it odd that it should be dedicated to yet another refutation.

We will be better able to understand the issue to be addressed in this chapter, if we return to the aporia at 315b20-3. We recall the first horn broadcasting impossible consequences to follow if generation is association, and identified the ontology on which this account rests as such a consequence. Moving on to the second horn, we read of compelling arguments that generation cannot be happening in any other way but by association. These compelling arguments obviously refer to the upcoming reductio. Prima facie, the warning issued in the second horn seems to be that these compelling arguments threaten to revive atomism. But the sequel to that horn, offered in the immediately following sentence, falsifies this reading. “If generation is not association, there is no generation or it is alteration.”9 Two things call for our attention here: First, if the compelling arguments just mentioned reinstate atomism, the suggestion in the sequel that generation might not be association is entirely out of place. For, generation as association is the atomist position, and if the compelling arguments restore the latter, they also affirm the former. Second, it is unclear that there should be no generation at all, if generation were not association. The case should be rather the opposite. If generation is not association, then so much the better for the forthcoming Aristotelian account, according to which there is generation and it is not association.

In fact, the atomist account of generation and corruption is hardly a competitor to Aristotle’s, even when it is seen apart from the ontology it presupposes. Aristotle indicates as much later in the chapter. At 317a17-20 he implies that he has been too generous to recognize association and generation as accounts of generation and corruption. Properly speaking they are not. For Aristotle, generation and corruption are radical changes as they occur at the level of being: generation creates substances, corruption destroys them. Alteration, on the other hand, depends on existing substances. However, Metaphysics Z, identifies the atoms as the Democritean substances and applauds Democritus for making substantiality the reason for holding that they are indivisible: “therefore if substance is one, it will not consist of

9Here is the text of the aporia and the sequel. ‘Ἀπορίας γὰρ ἔχει ταῦτα καὶ πολλὰς καὶ εὐλόγους. Εἰ µὲν γάρ ἐστι σύγκρισις ἡ γένεσις, πολλὰ ἀδύνατα συµβαίνει· εἰσὶ δ' αὖ λόγοι ἕτεροι ἀναγκαστικοὶ καὶ οὐκ εὔποροι διαλύειν ὡς οὐκ ἐνδέχεται ἄλλως ἔχειν. Εἴτε µή ἐστι σύγκρισις ἡ γένεσις, ἢ ὅλως οὐκ ἔστι γένεσις ἢ ἀλλοίωσις,’ (315b19- 3). “For they lead to both many and well-grounded aporiai. On the one hand, if generation is association, many impossible consequences follow. On the other, there are other compelling and not easy to unravel arguments that generation cannot take place in any other way. If generation is not association, either there is no generation at all, or it is alteration.”

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substances present in it, also according to the argument which Democritus states rightly; because it is impossible, he says, that one should come to be from two or two from one, since he identifies his indivisible magnitudes with substances.”10 Unlike the Aristotelian substances, Democritus’ are indestructible and eternal. They underlie and survive all change, which makes association and dissociation—generation and corruption according to Democritus, alteration. Democritus does not account for generation or corruption. He reduces them to alteration. Aristotle does not therefore need to refute the atomist ontology to justify that its account of generation is poor and that an adequate alternative is needed.

On Aristotle’s account, we have generation when form informs some portion of matter to bring a substance into existence. A prerequisite for matter to take on form is that it is divisible. If matter is divisible so is also body.11 Divisibility of body is essential also to the accounts of the physical processes examined immediately after generation, corruption and alteration, namely growth (αὔξησις) or diminution (φθίσις) of natural substances, as well as µίξις—to be rendered here as ‘fusion’.12 Growth and diminution are of homoeomerous parts such as bone, blood, flesh, hair or plant tissues and depends on fusion. In fusion the bodily components combining are preserved potentially but not actually, which they can do only if they are divisible.13

The divisibility thesis must be secured, if GC is to go ahead with the announced program. For, compelling other arguments (λόγοι ἕτεροι) raise an obstacle to that thesis. These arguments are said to be ‘other’ for the reason that they have little to do with resuscitating atomism. As we have seen, both the first horn of the aporia at 315b20-3 and the sequel to the second are adamant that atomism is impossible. That is what takes its account of generation off the table.

10 ὥστ’ εἰ ἡ οὐσία ἕν, οὐκ ἔσται ἐξ οὐσιῶν ἐνυπαρχουσῶν καὶ κατὰ τοῦτον τὸν τρόπον, ὃν λέγει Δηµόκριτος ὀρθῶς· ἀδύνατον γὰρ εἶναί φησιν ἐκ δύο ἓν ἢ ἐξ ἑνὸς δύο γενέσθαι· τὰ γὰρ µεγέθη τὰ ἄτοµα τὰς οὐσίας ποιεῖ. (1039a7-11). 11 A simple body is matter qualified by qualities belonging to two sets of contraries: warm /cold and wet /dry. Changes in the qualities that qualify matter lead to the transformation of a simple body into another. The qualities qualifying matter do not turn divisible matter into indivisible body. If body is indivisible, so is matter. 12 Aὔξησις and φθίσις is discussed in I.5, µίξις in I.6. Vasilis Kalfas (2011), 11-33 argues that a chief aim of GC is to offer detailed accounts for claims regarding physical changes and processes Aristotle mentions but does not explain in detail in the Physics. Hence also the frequent, implicit or explicit, references to the Physics in GC. Apart from the ones mentioned above, accounts of important physical processes include contact between bodies as a precondition of active and passive interaction (ποιεῖν and πάσχειν) and the movements of the heavenly bodies as a cause of regularities in the sublunary region as for instance the eternal cycle of generation and corruption. 13See also the discussion on the production of colors where it is claimed that mixtures of distinct indivisible smallest units produce compounds of discrete units, whereas a mixture by complete fusion (ὅλως πάντῃ πάντως) is only possible with divisible ingredients (De Sensu 440a39-b12).

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So, atomism is unlikely to be the obstacle that needs to be removed, and that is not only because both horns of the aporia declare it to be impossible. More importantly, a rehabilitation of atomism would give us at least two, albeit conflicting, accounts of generation: the atomist account and Aristotle’s forthcoming one. What Aristotle says is that failure to resolve the aporia would leave us with no account of generation. So we have a challenge ahead of us in identifying the obstacle raised by the compelling arguments.

II Before doing so, however, we need to examine the possibility, still left wide open, that the compelling arguments identify a Democritean move against divisibility. Even though atomism is already refuted, there may still be an argument by Democritus that threatens the divisibility thesis.

All the accounts of the processes for which GC I.2 is to clear the ground depend on the divisibility of body. But the compelling arguments threatening Aristotle’s prospective account of generation and corruption concern divisibility of magnitude, not just body. ‘Σῶµα’ and ‘µέγεθος’ are used interchangeably, throughout.14 Such use is standard Aristotelian practice. In Phys. III.5 (204b6-7), both ‘magnitude’ and ‘body’ are used to refer to either one of these two. So, we are told of two kinds of body, sensible (αἰσθητόν) and intelligible (νοητόν), both of which are said to be magnitude. Intelligible body is that of the mathematical entities. As body, it too has matter, namely intelligible matter (ὕλη νοητή), which we are told in the Metaphysics is the matter of the sensibles when they are not considered as sensibles.15 In using ‘body’ and ‘magnitude’ interchangeably, GC I.2 follows a practice well established in other Aristotelian works. On matters concerning divisibility, Aristotle does not distinguish between body and magnitude. Whatever is true of the one is also true of the other.

We get a glimpse of the theoretical basis for this in Phys. III.7. Aristotle there instructs the mathematicians not to allow in their study the infinite in the direction of addition, for if they did, they would be risking that the magnitude they study surpasses the existing magnitude (207b27-34). That would amount to abandoning geometry’s proper object of study, which is

14Twice we get “σῶµα καὶ µέγεθος” (316a15, 316b15), but this is merely an ἕν διὰ δυοῖν construction. See also Luria (2007), 1016, 2. The correct rendering of this clause is “body, that is to say magnitude”. Joachim’s translation of “σῶµά τι εἶναι καὶ µέγεθος πάντῃ διαιρετόν” at 316a15: “body (i.e. a magnitude) is divisible everywhere”. 15 νοητὴ δὲ ἡ ἐν τοῖς αἰσθητοῖς ὑπάρχουσα µὴ ᾗ αἰσθητά, οἶον τὰ µαθηµατικά (Met. 1036a11-2)

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body in nature. That is what geometry studies: body in nature albeit in an idealized form. It studies body merely as magnitude, i.e. as a spatial expanse the essential attribute of which is quantity. On this conception, the student of geometry studies sensible body stripped off its sensible attributes; he/she studies it as intelligible body, for intelligible body is just that: sensible body considered merely as magnitude, i.e. a spatial expanse of a certain quantity. To call it intelligible is not to imply that there is something else, over and above sensible body, graspable only by some intellectual, non sense-based epistemic faculty. It is intelligible because thought reveals it to itself by stripping sensible body, the only body that is, off its sensible attributes. If geometry teaches that body thus considered is divisible, that is because sensible body is. For the attributes that make it sensible have zero bearing on whether body is divisible or not.

For simplicity, though anachronistically, let us use ‘sensible body’ and ‘intelligible body’ also when talking about Democritus, as follows: ‘sensible body’ for what he calls ‘body’ and ‘intelligible body’ simply for atoms considered as geometrical solids. Intelligible body divisibility for Democritus would then be that parts can be marked in an atom and atoms considered as geometrical solids are divisible. Now if the reductio in GC I.2 is a Democritean refutation of divisibility, we should take the Democritean view to be that both sensible and intelligible body thus specified is indivisible.16 It is uncontested that Democritus claims indivisibility of the former. But do we have evidence other than GC I.2, that he also claims indivisibility of intelligible body?17 For, we do, surely, need strong evidence to saddle Democritus with a view that would prevent him from distinguishing, say, the part of the O- shaped atom adjacent to an N-shaped one from its part that is adjacent to the T-shaped atom.

16Though Barnes and Sedley attribute to Democritus indivisibility of sensible body only, they still hold him responsible for this argument. They pass in silence the difficulty that this attribution saddles Democritus with an argument for the indivisibility of both sensible and intelligible body. 17There is plenty of clear evidence that Democritus did not claim indivisibility of intelligible magnitude. Speaking on the many senses of ‘indivisible’ in his commentary on Physics I.2, Simplicius notes that Democritus’ atoms have size and parts but are incapable of being divided because of their solidity and fullness. (Εἰ δὲ οὕτως ἓν τὸ ὂν ὡς ἀδιαίρετον, ἐπεὶ τὸ ἀδιαίρετον πολλαχῶς, ἢ τὸ µήπω διῃρηµένον οἷόν τε δὲ διαιρεθῆναι καθάπερ ἕκαστον τῶν συνεχῶν, ἢ τὸ µηδὲ ὅλως πεφυκὸς διαιρεῖσθαι τῷ µὴ ἔχειν µέρη εἰς ἃ <ἂν> διαιρεθῇ, ὥσπερ στιγµὴ καὶ µονάς, ἢ τῷ µόρια µὲν ἔχειν καὶ µέγεθος, ἀπαθὲς δὲ εἶναι διὰ στερρότητα καὶ ναστότητα, καθάπερ ἑκάστη τῶν Δηµοκρίτου ἀτόµων. 81.34-82.3). At 140.6-18 he quotes Porphyry’s account of Xenocrates’ response to a dichotomy argument (more on this later), according to which there exist indivisible magnitudes (µεγέθη), namely lines. There is not a word by either Porphyry or Simplicius that Democritus may have held a similar view. Also Scholia in Arist. 469b14 (Brandis): τῶν ἄτοµα φησάντων οἱ µὲν ἄτοµα σώµατα δοξάζουσιν, ὡς Λεύκιππος καὶ Δηµόκριτος, οἱ δὲ ἀτόµους γραµµάς ὡς Ξενοκράτης. (Of those who speak of atoms, some think that there are indivisible bodies, as do Leucippus and Democritus, others that there are indivisible lines, as does Xenocrates).

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And given that there are infinitely many atoms in as many shapes and sizes, he ought surely to allow himself to say of any one of them that it is many times bigger or smaller than another.

Elsewhere in the GC, Aristotle implies that Democritus and Leucippus claim indivisibility of sensible body, though not of intelligible body. 327a8 distinguishes between two positions, one that claims indivisibility of body and one of surface (πλάτος). Of these, Aristotle attributes to Democritus and Leucippus the first and implies that they do not hold the second. He repeats this in our chapter when he asks at 315b28-30: “If there exist [indivisible] magnitudes, are these bodies as claims Democritus and Leucippus, or planes as claims Plato in the Timaeus?”18

Even so, other remarks Aristotle makes have been thought to be evidence that he interprets Democritus as claiming intelligible body to be indivisible. In the De Caelo, commenting on Plato’s view that the elements dissolve into triangles that can then re-combine to compose other elements, he says: “Besides this they are compelled to assert that not every body is divisible, and in that way hold a physical doctrine that is out of step with the most exact science. For mathematics take also the intelligible body to be divisible, whereas those will not accept that even the sensible is divisible in order to save their theory.” Then Aristotle seemingly generalizes the charge with a remark that appears to some to be targeting atomists of all persuasions: “necessarily all those who assign a shape to each of the elements and determine their essence (τὰς οὐσίας αὐτῶν) in terms of shape, they make them indivisible.”19 First thing we notice is that the target here are not the Atomists. Whoever is targeted is said to hold some sensible body to be indivisible on account of shape, and such a charge against the Atomists would conflict with what Aristotle says about them elsewhere. For, though the shapes of their atoms are unalterable, they are because they are indivisible. They are not indivisible on account of their shape.20

18 εἰ µεγέθη, πότερον, ὡς Δηµόκριτος καὶ Λεύκιππος, σώµατα ταῦτ' ἐστίν, ἢ ὥσπερ ἐν τῷ Τιµαίῳ ἐπίπεδα; 19 Πρὸς δὲ τούτοις ἀνάγκη µὴ πᾶν σῶµα λέγειν διαιρετόν, ἀλλὰ µάχεσθαι ταῖς ἀκριβεστάταις ἐπιστήµαις· αἱ µὲν γὰρ καὶ τὸ νοητὸν λαµβάνουσι διαιρετόν, αἱ µαθηµατικαί, οἱ δὲ οὐδὲ τὸ αἰσθητὸν ἅπαν συγχωροῦσι διὰ τὸ βούλεσθαι σῴζειν τὴν ὑπόθεσιν. Ἀνάγκη γὰρ ὅσοι σχῆµα ποιοῦσιν ἑκάστου τῶν στοιχείων καὶ τούτῳ διορίζουσι τὰς οὐσίας αὐτῶν, ἀδιαίρετα ποιεῖν αὐτά· (De Caelo. 306a26-32) 20 It is true that, on the evidence of the Timaeus, Plato takes two specific triangles to be the ultimate elemental constituent, but whether this commits him to indivisibility of sensible or intelligible body is an issue far too complex to deal with here.

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Second, the charge, obviously against Plato, is that he goes against mathematics because he holds that not all body is indivisible.21 There are two things to notice here. First, the idea seems to be that sensible body is evidently divisible and that intelligible body is too, therefore. Consequently, Aristotle does not say that Plato claims indivisibility of intelligible body. At most, he implies that Plato is committed to it by holding sensible body to be indivisible. Indeed, it is Plato’s view on body that conflicts with mathematics, and that Aristotle says, I suggest, on the basis of his own view on mathematics. For Aristotle geometry is the study of intelligible body, which is sensible body considered as spatial expanse with quantity.

In a similar statement earlier in the De Caelo Aristotle does target the Atomists: “moreover, they hold a physical doctrine that is out of step with mathematics when they say that there are indivisible bodies. And they cancel many of the common opinions and what is apparent to the senses, as we have said previously in the works on time and motion.”22 First thing we notice is that when Aristotle mentions what he takes the Atomists actually to be claiming, he only mentions indivisibility of sensible body. He then charges that they are thereby committed to two unwelcome consequences: (a) they are going against mathematics; (b) they are out of step with evidence available to the senses. Charge (a), the interesting one for us, expresses Aristotle’s own views. For him, the only difference between intelligible and sensible body is that the former lacks, whereas the latter has, sensible attributes. But these attributes have no bearing on body as such, i.e. as specially extended quantity, other than making sensible the bodies that possess them. If geometry claims that body as such is divisible, it makes a claim about all body, and so if one holds indivisibility of sensible body, one goes against geometry.23

We can now turn to a remark by Simplicius that Epicurus departs from Democritus and Leucippus in holding the atoms not to be partless (ἀµερῆ). This is offered as evidence that

21 One might be tempted to take “not all body” to mean that there is some, possibly intelligible, body that Plato thinks is divisible. Most likely what Aristotle means is that Plato allows body to be divisible but only down to its constituent triangles. 22 Πρὸς δὲ τούτοις ἀνάγκη µάχεσθαι ταῖς µαθηµατικαῖς ἐπιστήµαις ἄτοµα σώµατα λέγοντας, καὶ πολλὰ τῶν ἐνδόξων καὶ τῶν φαινοµένων κατὰ τὴν αἴσθησιν ἀναιρεῖν, περὶ ὧν εἴρηται πρότερον ἐν τοῖς περὶ χρόνου καὶ κινήσεως. (De Caelo 303a20-4). 23 That is exactly what Aristotle confirms when he writes: “for sidestepping the truth even in the slightest becomes the cause of ten-thousandfold deviations later on. As in the case of someone saying there is some minimum magnitude. For, by introducing this minimum he would be shaking the foundations of all mathematics.” (εἴπερ καὶ τὸ µικρὸν παραβῆναι τῆς ἀληθείας ἀφισταµένοις γίνεται πόρρω µυριοπλάσιον. Οἷον εἴ τις ἐλάχιστον εἶναί τι φαίη µέγεθος· οὗτος γὰρ τοὐλάχιστον εἰσαγαγὼν τὰ µέγιστ' ἂν κινήσειε τῶν µαθηµατικῶν. Ibid. 271b8- 11). Just by introducing some indivisible magnitude, it does not matter of what kind that is, one would be upsetting the whole of mathematics.

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Democritus and Leucippus deny divisibility of intelligible body.24 If anything, properly understood this passage confirms that they affirm it.

Given that, of the things that are, five are magnitudes, i.e. line, surface, body and moreover motion and time, a notion everyone shares is that each one is comprised of parts to which it is also divided. Individually, however, some hold that whichever magnitude one takes is divided in magnitudes ad infinitum, so that the division never results in partless things and for this reason they say that the magnitudes are comprised of parts and not of partless things. The others, desperately rejecting ad infinitum cutting, as we are not able to cut to infinity and in that way be convinced of the incompletability of cutting, said that bodies consist of indivisibles and are divided into indivisibles; but while Leucippus and Democritus think that not only impassivity is the cause of the primary bodies’ not being divided but also that they are small and partless, Epicurus later did not hold that they are partless, but said that they are indivisible due to their impassivity.25

Reporting first on Aristotelian doctrine, Simplicius speaks of five kinds of magnitude: line, surface, body, time and movement. Aristotle reserves the term ‘magnitude’ for the first three,26 which is why Simplicius subtly sets them apart from motion and time. Geometrical solids too are divisible on this view, but are not mentioned. In a report on the Aristotelian view, as is this, the mere mention of body is sufficient: both it and geometrical solids are magnitude. But notice that when Simplicius turns to the Atomists he says, pointedly, that they claim not magnitude, but body to be indivisible. This would be peculiar, if he thought the Atomists claim that what Aristotle calls magnitude is indivisible. In fact, the case seems rather to be that Simplicius mentions one by one all three kinds of spatially extended magnitude (i.e. in one, two, and three dimensions) instead of following Aristotelian practice to use ‘magnitude’ to refer to all in order to identify the one the Atomists claim is indivisible. So he refers to the atom as ‘primary body’ (πρῶτον σῶµα), and never speaks of primary magnitude, or, for that matter, primary line or surface.

24 See Furley (1967), 87 ff. 25Πέντε ὄντων ἐν τοῖς οὖσι µεγεθῶν, γραµµῆς ἐπιφανείας σώµατος καὶ ἔτι κινήσεως καὶ χρόνου, τὸ µὲν συγκεῖσθαι ἕκαστον ἐξ ἐκείνων τῶν µερῶν εἰς ἃ καὶ διαιρεῖται, κοινή τις πᾶσιν ἔδοξεν ἔννοια. ἰδίᾳ δὲ λοιπὸν οἱ µὲν ἐπ’ ἄπειρον πᾶν τὸ ληφθὲν µέγεθος εἰς µεγέθη διαιρεῖσθαι νοµίζουσιν, ὡς µηδέποτε τῆς διαιρέσεως εἰς ἀµερῆ καταληγούσης, καὶ διὰ τοῦτο καὶ ἐκ µερῶν συγκεῖσθαι τὰ µεγέθη λέγουσι καὶ οὐκ ἐξ ἀµερῶν· οἱ δὲ τῆς ἐπ’ ἄπειρον τοµῆς ἀπεγνωκότες, ὡς οὐ δυναµένων ἡµῶν ἐπ’ ἄπειρον τεµεῖν καὶ ἐκ τούτου πιστώσασθαι τὸ ἀκατάληκτον τῆς τοµῆς, ἐξ ἀδιαιρέτων ἔλεγον ὑφεστάναι τὰ σώµατα καὶ εἰς ἀδιαίρετα διαιρεῖσθαι. πλὴν ὅτι Λεύκιππος µὲν καὶ Δηµόκριτος οὐ µόνον τὴν ἀπάθειαν αἰτίαν τοῖς πρώτοις σώµασι τοῦ µὴ διαιρεῖσθαι νοµίζουσιν, ἀλλὰ καὶ τὸ σµικρὸν καὶ ἀµερές, Ἐπίκουρος δὲ ὕστερον ἀµερῆ µὲν οὐχ ἡγεῖται, ἄτοµα δὲ αὐτὰ διὰ τὴν ἀπάθειαν εἶναί φησι (925.5-17) 26Phys. 207b 21-5; De Caelo 268a7-10

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The key term in this report on the Aristotelian account is ‘partless’ (ἀµερές). A part (µέρος) is of magnitude and one of several of which magnitude is said to be composed (συγκεῖσθαι). To the parts of which magnitude is composed it is also divided, and, on this view, any divided part can be severed from any other. If a part is partless, on the other hand, it is not composed of parts and parts can be severed from it. So, to say that magnitude on Aristotle’s view is not comprised of anything partless (οὐκ ἐξ ἀµερῶν) is to say that it is composed of divisible and severable parts.

Simplicius then says that Democritus holds the atoms to be indivisible (ἀδιαίρετα): one of the reasons is that they are partless. ‘Partless’ in this context can only mean what it means in the report on Aristotle: if parts cannot be severed from the atom, the atom is partless, and indeed we know that the Democritean atoms are not composed of parts that cannot be severed from them. So, there is no information here as to whether Democritus allows parts to be marked in an atom or not. The remark on Epicurus does not change this fact. Epicurus is mentioned at this juncture as representing a special case. Like Democritus he holds that atoms are indivisible: parts cannot be severed from them. But, unlike Democritus, he holds that atoms are nonetheless composed of parts. Therefore, partlessness is not available to Epicurus as a reason for the atoms to be indivisible, and he claims the reason to be impassivity instead.27

If anything, this passage is evidence that the Atomists assume divisibility of intelligible magnitude. Simplicius says they reject that body can be divided ad infinitum on the grounds that it is impossible to obtain empirical evidence that it can be. They could be thinking that such confirmation is impossible because body is comprised of indivisibles. But that would be begging the question. The reason Simplicius attributes to them instead is that such evidence would at some point have required performing an infinite number of divisions, which it is not possible to do: “as we are not able to cut to infinity” (οὐ δυναµένων ἡµῶν ἐπ’ ἄπειρον τεµεῖν).28 This is proof that, on Simplicius reading, the Atomists accept divisibility of intelligible magnitude. For they say not that it is impossible to divide body, but that it is impossible to be doing so forever. Dividing a body forever, which they rightly consider non-

27 Notice also that Epicurus is not reported to have said that the atoms are divided into parts, which one could argue is damaging for the view defended here, as it could suggest that Simplicius uses the term ‘divide’ also for marking parts on the atom, and so he differs in that way too from Democritus. 28Barnes (1982) objects that the Atomists fallaciously argue that body cannot be divided ad infinitum on the grounds that there cannot be evidence that it can be (p. 350-1). But the Atomists cannot be so arguing, unless they make themselves, and all too obviously, vulnerable to Barnes’ objection, for they too deny that empirical evidence can be had for their view.

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realizable due the fact that the task in incompletable, entails actually marking parts in that body forever.29 So the evidence, other than GC I.2, adduced in favor of the view that the Atomists hold intelligible body to be indivisible fails to support it. If anything, it rather suggests that they only claim indivisibility of sensible body, which is in agreement with what Aristotle says about them in the GC.

III Might it be that Democritus proves more than he intends to prove, and believes he does?30 That he misguidedly constructs an argument for the indivisibility of magnitude, though he is only interested in the indivisibility of sensible body? It is time to look at the argument itself.

Since, therefore, the body is divisible everywhere, let it have been divided. What, then, will there be left? Magnitude? No: that is impossible, for then something will not have been divided, and it was supposed to be divisible everywhere. But then, if neither body nor magnitude will be left, though division is to take place, body will either be made out of points, and be composed of something without magnitude, or of absolutely nothing; so that it would come to be out of nothings, and be composed of nothings, and the whole thing will then presumably be nothing but an appearance. Similarly, if it is made out of points, it will not have quantity.31

The conclusion that dividing a magnitude turns it into nothing or magnitudeless (ἀµεγέθη) points only follows on the completability assumption, namely that dividing a magnitude everywhere is a completable task: at some point in time, the process of dividing a magnitude everywhere comes to an end in the sense that there will not be magnitude left to divide. We will ask later about possible grounds for making this assumption. For the moment we notice that the completability assumption contradicts what the Simplicius passage we examined

29 The interesting question is why the Atomists demand empirical confirmability to accept that body is divisible, when they hold their own view to be empirically non-confirmable. I suggest the reason is that it is incredible to them (hence the ἀπεγνωκότες) that body should be divisible. First, the idea seems counterintuitive. Significantly, Plato too thought that earth is the only element that is not transformable to another. Body in its sturdiness makes a firm claim to being, in their view, as it is prime candidate for meeting their chief metaphysical requirement, which is oneness. So, in addition to being offensive to intuition, divisibility of sensible body clashes with their core metaphysical belief to such an extent that nothing short of hard empirical evidence would make them consider it. 30 This is how C. C. W. Taylor (1999) makes compatible the claims that (1) Democritus claims indivisibility of body only and (2) this argument against the divisibility of all magnitude is by him, both of which claims Taylor endorses (170). 31 Ἐπεὶ τοίνυν πάντῃ τοιοῦτόν ἐστι τὸ σῶµα, διῃρήσθω. Τί οὖν ἔσται λοιπόν; µέγεθος; οὐ γὰρ οἷόν τε· ἔσται γάρ τι οὐ διῃρηµένον, ἦν δὲ πάντῃ διαιρετόν. Ἀλλὰ µὴν εἰ µηδὲν ἔσται σῶµα µηδὲ µέγεθος, διαίρεσις δ’ ἔσται, ἢ ἐκ στιγµῶν ἔσται, καὶ ἀµεγέθη ἐξ ὧν σύγκειται, ἢ οὐδὲν παντάπασιν, ὥστε κἂν γίνοιτο ἐκ µηδενὸς κἂν εἴη συγκείµενον, καὶ τὸ πᾶν δὴ οὐδὲν ἄλλ’ ἢ φαινόµενον. Ὁµοίως δὲ κἂν ᾖ ἐκ στιγµῶν, οὐκ ἔσται ποσόν. (316a23- 30)

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previously implies that the Atomists hold. The reason that passage attributes to the Atomists for rejecting divisibility is that performing all the divisions a divisible magnitude can sustain is a non-ending task. It is odd, on that evidence, that Democritus should have constructed an argument against divisibility premised on the completability assumption.

More problematically, the completability assumption is in conflict with another premise, explicitly stated and clearly needed for the argument to be a reductio. Nothing in the conclusion would be absurd, if it were allowed that division consumes magnitude. But this is not allowed. In fact it is explicitly stated, and used as a premise, that division does not alter the size of the magnitude it divides. “Since, when the body was divided into two or more parts, 32 the whole was not a bit smaller or bigger than it was before the division”. If division does not alter the size of the magnitude it divides, it is unclear why dividing a magnitude would ever result in there being no magnitude left to divide further.

In fact the argument itself does everything it can to broadcasts that it is flawed. It assumes, and we shall have more to say about this later, that ‘division’ and ‘point’ are co-referential.33 Point is explicitly said to be magnitudeless (ἀµέγεθος) and, therefore, have no size. But, if division is a point and point is magnitudeless, it is obviously possible for any magnitude, however small, to receive a point. This is proof that the task of dividing a magnitude through and through is incompletable. It is simply incredible, then, that Democritus, or anyone else, should have constructed such a blatantly flawed argument in the belief that it could support atomism or refute the divisibility thesis.

It is often said that the reductio has ancient precedent in an argument often referred to as ‘dichotomy’.34 So maybe Democritus simply inherited the argument and neglected to reflect on its validity.35 One of the horns of this two-horned paradox, as Simplicius quotes it from

32Διαιρεθέντος γὰρ εἰς δύο καὶ πλείω, οὐδὲν ἔλαττον οὐδὲ µεῖζον τὸ πᾶν τοῦ πρότερον (316a31-3). So Aristotle accepts and assumes what also goes under the name ‘principle of complete additivity’, namely that the size of a magnitude partitioned into a set of parts is equal to that of the sum of the sizes of the set of parts. 33 In fact, the conclusion that the remainder would be nothing or without quantity rests on the thought that only the divisions would survives division and there is no quantity in division. 34 See Taylor (1999), 165 ff.; Skyrms, (1983), 225-6; Sorabji (1983) 336-41; Furley (1967), 84-5; Luria (1933), 107. 35 “The one from dichotomy” (ὁ ἐκ τῆς διχοτοµίας), says Aristotle in Phys. I.3 (187a 2-3). He also uses ‘διχοτοµία’ in Phys. VI.9 in the discussion of Zeno’s paradoxes about motion. Some hold that it is to those latter ones he refers with “the one from dichotomy (Furley (1967), 82). Fact is it is unclear whether Aristotle uses ‘διχοτοµία’ as a technical term to refer to some specific argument. I follow Simplicius who uses “dichotomy” as a catchall term for the paradoxes against plurality.

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Zeno, says that if Being is many (if there are many Beings), they are so small that they have no magnitude at all. This may be thought to be a precursor to the conclusion of the reductio in which division is assumed eventually to be turning magnitude to nothing. It is not. The aim of the paradox is to defend that only the One is and not the sensible many.36 In full, the it argues that if Being is not one but many, these many must either be so small that they have no 37 magnitude at all, or they must be infinite (139.7-8). Supposing that there are many things, each must be being; but it must also be one, for what is being is also one by doctrine. There is no sign anywhere that the conclusion of the first horn, namely that none of the many Beings has magnitude, is at all linked to the thought that division eventually annihilates magnitude. The conclusion is rather a consequence of the constraint that Being is “identical with itself and one” (ἑαυτῷ ταὐτὸν εἶναι καὶ ἕν).38 Each of the many sensibles has parts and is therefore not one but many. Ant it is not identical with any one of its parts. So it fails to be identical with itself, as it itself is one while its parts are many. To make these problems disappear the many must have no parts, and consequently no magnitude. But then the many too disappear. Each of the many must, therefore, have some magnitude, which now makes the problems reappear. For if each of the many has magnitude, each has parts, each one of which also has parts and so on with the parts of the parts. Both it and its parts are subject to divisions ad infinitum (ἐπ᾽ἄπειρον τοµή), and so each one of the many is not strictly one (ἀκριβῶς ἓν) but infinite, since it is infinitely many.39 Not only, then, does this paradox not mention or indicate anywhere that repeated divisions will eventually annihilate magnitude, but rather it is based on the premise that divisibility implies interminable divisions.

Simplicius then quotes an argument reported by Porphyry, which he also calls dichotomy.

This is markedly different from the one above. Purporting to show that “Being is only one and

36 Fragment. B 8,22. The Ancient commentators disagree among themselves whether the aim of Zeno’s paradox was to defend Parmenides’ One against the many sensibles or show that not even that is one. This debate has not bearing on my argument here. 37 εἰ πολλά ἐστιν, ἀνάγκη αὐτὰ µικρά τε εἶναι καὶ µεγάλα, µικρὰ µὲν ὥστε µὴ ἔχειν µέγεθος, µεγάλα δὲ ὥστε ἄπειρα εἶναι (141.6-8). Simplicius’ rendering of this same argument at 139.7-8 presents Zeno’s argument, differently than does the quote from Zeno. On Simplicius’ report, the second horn concludes that each of the many is infinite in magnitude, but ‘magnitude’ does not appear in the quote from Zeno. 38 139.16-9. Though the lines containing this argument in the Aldine edition differ from those in the edition of Diels, both texts agree with the interpretation offered here, as does also Simplicius who makes the same point: “none has magnitude, for the reason that each one of the many is identical with itself and one” (οὐδὲν ἔχει µέγεθος ἐκ τοῦ ἕκαστον τῶν πολλῶν ἑαυτῷ ταὐτὸν εἶναι καὶ ἕν. 139.17-8). 39 Themistius as quoted by Simplicius: “if it were divided, he says, it would not even be strictly one because of the cutting of bodies into infinity” (εἰ γὰρ διαιροῖτο, φησίν, οὐδὲ ἔσται ἀκριβῶς ἓν διὰ τὴν ἐπ’ ἄπειρον τοµὴν τῶν σωµάτων 139.21-2). Simplicius also quotes Alexander quoting Eudemus: “Zeno the companion of Parmenides tried to show that it is not possible for existing things to be many for none among them is one, and the many is a plurality of units” (99.13-8).

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this without parts and indivisible”,40 it argues that, if Being can be divided, either it is divided until it reaches infinitely many least indivisibles and the whole will then consist of them, or it will be divided until it disappears into nothing and will consist of nothing.41 Let us call this

‘dichotomy1’.

The origin of this argument is unclear. Porphyry credits it to Parmenides, but Simplicius righty complains that there is no evidence linking Parmenides to paradoxes of this kind. In the end Simplicius agrees with Alexander that it is by Zeno (140.18-26) and goes on to quote Zeno’s dichotomy as evidence (140.29-1.8). But the horns of Zeno’s argument, at which we just looked, state different conclusions (141.7-8). There is nothing in the quote from Zeno about infinitely many indivisible magnitudes or division dissolving magnitude into nothing as we have in the argument reported by Porphyry. Moreover, when Aristotle mentions in the Physics those who gave in to the argument from dichotomy, whomever he may be referring to,42 he indicates not that indivisible magnitudes were a possibility envisaged in one of its horns, but that the problems pointed to by either horn of the dichotomy led them to postulate indivisible magnitudes.

I rather suspect that the origin of the argument Porphyry reports is in fact GC 1.2. There is no independent evidence that Zeno’s dichotomy mentions indivisibles. Lack of such evidence might be why Porphyry wisely hesitates to attribute it to Zeno. But he nonetheless links it to the Eleatics presumably for a similar reason the moderns identify Zeno as the origin of the reductio in GC 1.2. This might be the apparent affinity between the conclusion of the reductio and the horn in Zeno’s dichotomy stating that the many have no magnitude. The reductio can be seen to offer an explanation for Zeno’s statement of that horn; and we have no explanation from Zeno, which makes it attractive to link the two. Be that as it may, the similarity between the horn of dichotomy1 and the conclusion of the reductio in GC I.2 is only apparent. The thing under division is the same in either horn. Since dividing it is said in the first horn to be resulting in infinitely many indivisibles, each one with magnitude of some size, we may safely assume that the magnitude of the thing under division is infinite. The magnitude under division in GC I.2 is finite. This means that the horn in Porphyry’s report cannot be saying

40 τὸ ὂν ἓν εἶναι µόνον καὶ τοῦτο ἀµερὲς καὶ ἀδιαίρετον (139.27). 41 ὡς ἤτοι ὑποµενεῖ τινὰ ἔσχατα µεγέθη ἐλάχιστα καὶ ἄτοµα, πλήθει δὲ ἄπειρα, καὶ τὸ ὅλον ἐξ ἐλαχίστων, πλήθει δὲ ἀπείρων συστήσεται· ἢ φροῦδον ἔσται καὶ εἰς οὐθὲν ἔτι διαλυθήσεται καὶ ἐκ τοῦ µηδενὸς συστήσεται· ἅπερ ἄτοπα. (139.29-32) 42 And they are most likely to be residents of the Academy.

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that repeated divisions would annihilate magnitude. Being infinite, the thing under division is inexhaustible and hence capable of sustaining an infinite number of divisions, even if each division were to be consuming some of it.

It is to the credit of Simplicius that he does not attribute the horn’s claim of disappearance of a magnitude under division to a corruptive effect of division itself. Instead, the lesson he expects the Atomists would have drawn, if confronted with dichotomy1, is the one he says 43 Xenocrates draws: If divisible, Being would dissolve in, and be consumed by, non-Being. This is the thinking of the atomists of all persuasions. The divisibility option is unacceptable to them because any part divided off any magnitude can itself have parts divided off it, and so on ad infinitum. No magnitude would then be one. This consideration, and nothing resembling the reductio, is the motivation for classical atomism Aristotle diagnoses in GC I.8. The Atomists, he says there, adopt the Eleatic principle that if everything is “divisible through and through, nothing is one”.44 Indeed, Aristotle’s report on Leucippus’ in the same chapter offers the Atomist response to Porphyry’s report: They opt for indivisibility because “out of what is truly one, no plurality can come to be nor does one out of what truly are many”.45

In the previous section we saw evidence in Simplicius that the Atomists being aware of the impossibility ever to perform all the divisions to which a divisible magnitude is subject. Here we found no precedent for the claim that divisions might eventually annihilate magnitude in Zeno’s dichotomy. Zeno’s dichotomy presupposes ad infinitum division, which contradicts the completability premise of the reductio in GC I.2. Were we to assume that Democritus authored the reductio to use it against the divisibility thesis, we would be assuming not only that he is responsible for a blatantly flawed argument, but also that he is blind to a tradition that correctly understands divisibility of magnitude as implying endless cuttings.

IV If the reductio is so obviously flawed, why does Aristotle dedicate a whole chapter to it rather than point out the falsity on which it rests? This question is more pressing now that we know that the reductio has no history. Why does he even bother to mention it, when everyone knows that no divisible magnitude can be exhaustively divided? Aristotle himself, explicitly

43 It is to this argument Xenocrates is said to be responding with indivisible lines. And, interestingly, though he opts for indivisible lines, he allows divisibility of magnitude. See 140.13-8. 44 Εἰ µὲν γὰρ πάντῃ διαιρετόν, οὐδὲν εἶναι ἕν (325a7-8) 45 ἐκ δὲ τοῦ κατ' ἀλήθειαν ἑνὸς οὐκ ἂν γενέσθαι πλῆθος, οὐδ' ἐκ τῶν ἀληθῶς πολλῶν ἕν (325a34-6).

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and unproblematically asserts in Phys. III.6 that a finite magnitude can sustain an infinite number of divisions that cannot all be performed. Exactly where is the completability assumption coming from?

In Phys. III.6, divisibility of magnitude is offered as grounds for the claim that the infinite (ἄπειρον) exists: the infinite must necessarily exist, for if it did not, “magnitudes would not be divisible in magnitudes”.46 “Magnitudes are divisible in magnitudes’’ is Aristotle’s way of saying that they are divisible—they are comprised of non-partless parts in Simplicius’ terminology. Phys. III.6 takes it for granted that magnitude can be subject to infinitely many divisions, and asserts on this basis that the infinite is in magnitude by division δυνάµει; ‘δυνάµει’ is here used in the special sense explained in Met. Θ.6 to indicate that the infinite is in magnitude never in actuality but only in the sense that magnitude can always be divided more times than however many times it has already been divided.

On the face of it, it would seem that Phys. III.6 has the answer to the reductio. A divisible magnitude can sustain infinitely many divisions, thus making the completability assumption false. Still, GC I.2 does not ever propose to solve the reductio in this way, and commentators wonder why this apparently obvious solution is overlooked. The answer, I suggest, is that it is not the solution to the problem Aristotle puts to himself. In Phys. III.6, the infinite is said to be a κατὰ συµβεβηκός, an attribute of magnitude.47 What attribute qualifies what subject is a matter that pertains to the subject, not to the attribute. So one is entitled to ask what it is about the subject48 magnitude that makes it true that the infinite is an attribute of it in the way Phys. III.6 claims. What makes it true that magnitude can be subject to divisions ad infinitum? Phys. III.6 never raises that question, and never answers it.49 GC I.2 contains the answer. It can be found in a carefully crafted preamble to the reductio that introduces a claim about magnitude, which, if true, would ground the position that the infinite is an attribute of it.

An aporia arises if one supposes that some body and magnitude is divisible everywhere, and that this is possible. For what will there be that escapes the division? If it is divisible everywhere, and if this is

46καὶ τὰ µεγέθη οὐ διαιρετὰ εἰς µεγέθη (206a11) 47 Phys. 204a9-12, 18-9, 29-30 48 That magnitude, and more precisely magnitude as found in sensible bodies, is the subject is explicitly asserted in Phys. III.7: “subject properly speaking is that which continuous and sensible” (τὸ δὲ καθ' αὑτὸ ὑποκείµενον τὸ συνεχὲς καὶ αἰσθητόν. 208b1-2). 49 Lear (1980) is exactly right when he writes that the infinite exists in magnitude “because of the structure of the magnitude” (193). Quite reasonably, he does not say what that structure might be, as this is not discussed in the Phys.

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possible, then it might, at one and the same moment, be at a state where it has been divided everywhere, even if the divisions had not been effected at one and the same time; and if this were to take place, it would involve nothing impossible. And, in the same way (i.e. at one and the same time), whether by bisection or whatever other manner, given that it is in its nature to be divisible everywhere, if it is actually divided, nothing impossible will have taken place; since, even if it has been divided ten thousand times, nothing impossible will have resulted, though perhaps nobody in fact would so divide it.50

We notice that the terms ‘possible’ (δυνατόν), ‘simultaneously’ (ἅµα) and ‘divisible everywhere’ (πάντῃ διαιρετόν) link what is here referred to as aporia to the aporia later on at 316b19-25, which states the issue with which this chapter deals. ‘Everywhere’ (πάντῃ), which qualifies ‘divisible’ both here and throughout the GC,51 marks the contrast with the indivisibility thesis, which says magnitude can be divided only at the joins of the indivisible parts composing it.52

That magnitude is divisible everywhere is said to imply that it is possible (δυνατόν) for it to sustain all the divisions it is subject to ‘at one and the same moment’, simultaneously (ἅµα). Aristotle is adamant on simultaneity. He repeats twice in the course of seven lines that it should be possible (δυνατόν) for it to be so divided and thrice on top of that that there should be nothing impossible (ἀδύνατον) in so dividing it it. Simultaneity is so important that Aristotle even warns the reader not to be distracted by the magnitude of the task and think it is impossible. Even if it is not in fact divided everywhere simultaneously, it should be possible for it to be so divided.

50 Ἔχει γὰρ ἀπορίαν, εἴ τις θείη σῶµά τι εἶναι καὶ µέγεθος πάντῃ διαιρε-τόν, καὶ τοῦτο δυνατόν. Τί γὰρ ἔσται ὅπερ τὴν διαίρεσιν διαφεύγει; εἰ γὰρ πάντῃ διαιρετόν, καὶ τοῦτο δυνατόν, κἂν ἅµα εἴη τοῦτο πάντῃ διῃρηµένον, καὶ εἰ µὴ ἅµα διῄρηται· κἂν εἰ τοῦτο γένοιτο, οὐδὲν ἂν εἴη ἀδύνατον. Οὐκοῦν καὶ κατὰ τὸ µέσον ὡσαύτως, καὶ ὅλως δέ, εἰ πάντῃ πέφυκε διαιρετόν, ἂν διαιρεθῇ, οὐδὲν ἔσται ἀδύνατον γεγονός, ἐπεὶ οὐδ’ ἂν εἰς µυρία µυριάκις διῃρηµένα ᾖ, οὐδὲν ἀδύνατον· καίτοι ἴσως οὐδεὶς ἂν διέλοι (316a14-23). (At 316a22 where I follow the MS reading instead of Joachim’s διῃρηµένα (διαιρεθ)ῇ. (See also Sedley (2004) p. 68, n. 8.) 51 Interestingly, ‘everywhere’ qualifies also divisible also throughout the De Caelo. 52 Commentators attributing the reductio to Democritus read the division envisaged here to be exclusively of the sort involving spatial separation of parts from other parts. Following Barnes (1979), Sedley thinks that we have “some substantive separation of its parts”, by which he means spatial separation; “otherwise there would be little motivation for the closing words”, these being “although perhaps no one would divide it” (68-9). But the closing words are as appropriate also if the division envisaged merely distinguishes parts in a magnitude. Doing this everywhere is as daunting as spatially separating parts. The closing words merely indicate that the argument does not depend on anyone’s actually dividing a magnitude, and nothing in the preamble indicates that the division is to be thought of in one way rather than the other. This is in line with Aristotle’s view that if it is possible mentally to distinguish parts in a magnitude it is also possible spatially to sever them from it.

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It could hardly be put more clearly that we are here to recall the Metaphysics Θ.3 claim on capacities as belonging to subjects:53 “and a thing is capable (δυνατόν) if there is nothing impossible (ἀδύνατον) in its having the actuality (ἐνέργεια) of that of which it is said to have the capacity (δύναµιν)”.54 A proposition to the effect that some subject a possesses capacity (δύναµις) f, the actualization of which would result in a’s coming to have attribute F, implies that a is capable (δυνατόν) of coming to be F(a) actually, by way of having f actualized. The talk here is not about just any possibility regarding a, but quite precisely the possibility of a having actualized capacity f that it possesses. Of any subject x having capacity f, by actualization of which it becomes F, we shall be saying ‘x is F(x) in capacity’.

That some a is F(a) in capacity, we are told further, is to say that there is nothing impossible (ἀδύνατον) in its actually coming to be F(a) through actualization of f: “there is nothing impossible (ἀδύνατον) in its having the actuality (ἐνέργεια) of that of which it is said to have the capacity (δύναµιν)”. Since a’s possessing f implies the possibility of the actualization of f in a, and a’s actually being F(a), if it is impossible for a ever to be F(a) we get, by modus tollens, that a does not possess f.55 If it is impossible for a to come to be F(a), then f is not a capacity of a, and it is false that a is F(a) in capacity.56 In addition to specifying what it is for a subject to be capable of being qualified in a certain way in virtue of possessing a capacity, this formula specifies a procedure for testing claims to the effect that a subject possesses a capacity.

53 By ‘subject’ here I mean that to which Aristotle refers with ‘ὑποκείµενον’. This can be one of two things: (1) a substance, as for instance ‘Socrates’, which can underlie attributes such as those referred to by ‘white’, or ‘learned’; (2) matter, or body, which can underlie form or qualities. See Met. Θ 1049a29-36. 54 ἔστι δὲ δυνατὸν τοῦτο ᾧ ἐὰν ὑπάρξῃ ἡ ἐνέργεια οὗ λέγεται ἔχειν τὴν δύναµιν, οὐθὲν ἔσται ἀδύνατον (Met. 1047a24-6). 55 This non-impossibility requirement entails a thick, often unnoticed, commitment. Suppose that my hair’s turning white at seventy and staying the color it is now are not co-possible. Even so, there is a sense of ‘possible’ according to which it is true both that it is now possible that my hair will turn white when I am seventy and that it is now possible that it will remain the color it is now. The kind of possibility implied by a capacity in a subject is of a different sort. Since white-haired at seventy and brown-haired at seventy are not co-possible, I cannot be white-haired in capacity and never get white hair at seventy or later—assuming I or anything else does not intervene to keep my hair from turning white; and so it is not true of me that I am white-haired at seventy, in capacity. See also Aristotle’s proof that what is eternal cannot be perishable in capacity, De Caelo I.12, 281b18- 23. 56 According to Malink & Rosen (2012), the inference “it is possible for the magnitude to have been divided everywhere” does not follow from the claim “some magnitude is divisible everywhere”, “as it is clear from the analogy of winning a game: typically, for every player it is possible that he or she wins, but it is not possible that every player wins” (27-8). This objection seems to me to be missing the target for two counts. First, Aristotle never says that the former claim simply follows from the latter; what he says, I suggest, is that it follows from it given the commitment implied by the claim that magnitude is divisible everywhere in capacity. Clearly there is a mistake here somewhere, but it is not the one pointed to by Malink & Rosen. Second, the reason it is not possible for every player to win the game—even though it is possible for each player to win it—is that it is not co-possible that every player wins it. But any division a magnitude is subjected to is co-possible with any other division to which it is also subject.

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This preamble announces the claim that magnitude is divisible in capacity. That is why being divisible everywhere is said to be in magnitude’s nature (πέφυκε) and to imply commitments that match exactly those on capacities specified in Met. Θ.3. The simultaneously constraint is the clincher: all the divisions to which magnitude is subject can be performed (ἅµα).57 Aristotle goes out of his way to stress that the ἅµα belongs, and he repeats it later in the aporia at 316b19-27. He insists on it because it implies completability. If all the divisions can be performed simultaneously, performing them all is a completable task. And completability is central because it appears to be required by the claim that magnitude is divisible in capacity. As we have seen, for any subject x, x’s being F in capacity implies that x can be F actually.

Cutting up a magnitude indefinitely does not meet the Met. Θ3 constraint on capacities. The change that brings a subject from being something in capacity to being that thing in actuality is such that it aims at an end it is destined to reach and will do so unless prevented. Reaching the end for which it aims implies that the subject no longer is in capacity what it now is in actuality.58 The infinite is an attribute of magnitude because magnitude is divisible in capacity, and the simultaneity constraint is inserted to ensure that the reader does not slip into thinking that all we get here is a restatement of the Phys. III.6 position. GC I.2 is intent on taking the further step of grounding that position. The infinite is an attribute of magnitude because magnitude is divisible in capacity. But as it now turns out, the claim that magnitude is divisible in capacity appears to imply the possibility of an exhaustive division.

57 It is easy to think that this simultaneity requirement is the mistake in the argument eliminating which solves the reductio. But Aristotle shows no inclination to address the reductio by eliminating it. Nonetheless, commentators universally believe this to be the solution. Joachim actually claims that this is the solution offered in the text (1922; p. 77)—though it is nowhere there to be found. Williams (1982; 73-9) and Sedley (2004; 78, n. 28) say that it ought to be but Aristotle fails or does not explicitly offer it. Betegh (2006) too thinks that the reductio rests on an illicit transition from “(a) a body is divisible at every point” (every point = πάντῃ) to “(b) it is possible that the body is simultaneously divided at every point”. He thinks it is peculiar that Aristotle does not reject (b), since “according to the well-known Aristotelian doctrine, a physical or mathematical magnitude is potentially infinitely divisible” (276), so that “it is impossible simultaneously to actualize the divisions at every point”. He attributes this peculiarity to Aristotle’s being “charitable enough to grant this to Democritus” (277). But we have zero reason to believe that Democritus would have committed this grave fallacy, other than attributing this argument to him. Moreover, it is hardly charitable to grant the opponent a blatant fallacy, as Aristotle would be doing here, if Betegh is right. It makes even less sense to dedicate a chapter to dealing with an argument based on it, and never refute it by exposing it as the fallacy that it is. 58 Phys. 201b29-32; Met. 1048b29-25. Also DA 431a6-7: we have actuality unqualifiedly (ἁπλῶς ἐνέργεια) only when the actualizing movement reaches the end (τοῦ τετελεσµένου), and Met. 1050b6: atuality is prior to potency becausse it can, whereas potency cannot, exist without the other.

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V Without a coherent account of divisibility as a capacity of magnitude, the Phys. III.6 position is unfounded. Worse still, the reductio leaves it and the entire program of GC exposed. That explains why Aristotle sets off on the task of offering such an account and by gradually uncovering a further crucial but untold premise on which the reductio rests. Seemingly to avoid that the remainder of an exhaustive division is nothing, he proposes in the passage we saw in section III that it could be points. This is all too obviously a ploy. If something does survive, if something is left that cannot be divided further divisions, it is indivisible, which contradicts the divisibility hypothesis. This is sufficient reason for this suggestion never to have been made. To be sure, points in Aristotle’s view are indivisible because they are magnitudeless (ἀµεγέθη), as he calls them here. So, it might be said, the indivisibles implied by this suggestion are benign. But as Aristotle himself says, magnitude would then be composed (σύγκειται) of magnitudeless parts, and further down: “even if all points were put together they would not make any magnitude” (κἂν πᾶσαι συντεθῶσιν οὐδὲν ποιήσουσι µέγεθος 316a33-4). This ought to be sufficient reason to abandon this proposal immediately after it is made.

Instead of abandoning it, Aristotle goes on to offer a series of short arguments seemingly aiming at establishing that points are magnitudeless. But this is not what these arguments aim to do, surely. First, it is already asserted a couple of lines above that points are magnitudeless, and it has already been established elsewhere that they are.59 More importantly, whether points have magnitude or not is a theoretical question, not one to be decided on the basis of empirical arguments such as those we’ll soon visit. The aim of introducing points at this juncture and offering the short section of argument that follows is, I suggest, to alert the reader to a deeply ingrained foundationalist conception of magnitude as being composed of the points, against which conception Aristotle uses the opportunity to articulate his own. The frequent occurrence of the verbs συντίθηµι and σύγκειµαι, we shall soon see, is a clear proof of that.

59Understandably, commentators have had difficulties making sense of this argument. Joachim (1920), 79 barely mentions it, and Williams thinks that it is not meant seriously (1970), 70. Sedley’s (2004) claim that it is Aristotle’s attempt to capture a Democritean argument that points are without magnitude (p. 70) is incredible. We know nothing about any view Democritus may have had on point, and, as we are going to see, this argument does not actually argue that point is without magnitude.

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We understand Aristotle’s concern to broadcast his conception of point, in the context of a discussion on divisibility especially, once we appreciate his frustration with the treatment the notion receives by his contemporaries. In the Metaphysics he attacks Plato for holding that the point is a fiction of the mathematicians and charges Plato for being misled by a view that there exist indivisible lines.60 More interesting for present purposes are the people mentioned in Metaphysics M, probably heirs of Plato, who are said to hold that point generates magnitude and is like the one.61 On Aristotle’s view, that point generates magnitude implies commitment that it has itself magnitude and that it is like the one that it is indivisible. These may well be the same people Aristotle says in the De Anima hold that “a moving line generates a plane while a [moving] point a line”.62

In Aristotle’s view, point is something. In the Metaphysics he says it is indivisible and has position (1016b25).63 Having position entails being somewhere and hence being. Moreover, the motivation for the proposal that the remainder of a division might be points is to avoid that magnitude is made out of nothing (ἐκ µηδενὸς) and fails not because magnitude would then be made out of nothing but because it would be no quantity (οὐκ ἔσται ποσόν). Point is something for Aristotle, but it may be unclear what exactly it is. He unveils his conception in the following short section of argument.

Similarly, if it is made of points, it is no quantity. For when the points were in contact and coincided to form a single magnitude, they did not make the whole bigger. For, when it (the whole) was divided into two or more parts, it was not a smaller or bigger than it was previously, so that even if all of them

60 ἔτι αἱ στιγµαὶ ἐκ τίνος ἐνυπάρξουσιν; τούτῳ µὲν οὖν τῷ γένει καὶ διεµάχετο Πλάτων ὡς ὄντι γεωµετρικῷ δόγµατι, ἀλλ’ ἐκάλει ἀρχὴν γραµµῆς—τοῦτο δὲ πολλάκις ἐτίθει—τὰς ἀτόµους γραµµάς. (Met. 992a20-4). It is unlikely that Plato ever held the view Aristotle attributes to him here, though this is too large an issue to treat properly now. Suffice it to mention that if Plato holds there to be a Form of the Line, he could be allowing the magnitudes that mathematicians use in their studies, including lines to be divisible while also holding, plausibly, that the Line is indivisible. Points, on this view, would be useful to the mathematicians as divisions of the mathematical lines (τοµαί γραµµῆς). They would also be a fiction because there does not exist a Form of the Point. A Form of the point would be redundant, on such a view, for point is a division of a line and the Line cannot be divided. Aristotle’s objection that even the indivisible lines have a limit and that this implies the existence of points is off the mark, premised as it is on his own view. For him, a line-segment does indeed imply the existence of points because points are the limits that constitute it as such. Not so if (1) line is indivisible and hence (2) any larger line will only be divisible into its indivisible line-constituents. 61ἕτεροι δὲ [τὰ µεγέθη γεννῶσιν] ἐκ τῆς στιγµῆς (ἡ δὲ στιγµὴ αὐτοῖς δοκεῖ εἶναι οὐχ ἓν ἀλλ' οἷον τὸ ἕν. (Met. M 1085α31-3; also 1076b5-8 where Aristotle criticizes thinkers who treat point as a substance and existing separately). 62 κινηθεῖσαν γραµµὴν ἐπίπεδον ποιεῖν, στιγµὴν δὲ γραµµήν (409a4-6). Plutarch attributes this to Xenocrates in De Anima Procreatione 1012d, but it could as well be Speusippus. 63 [ἀδιαίρετον] πάντῃ καὶ θέσιν ἔχον (1016b25).

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be put together, they will not make any magnitude.64

If magnitude were made out of magnitudinous points that survive division, and each division corresponds to one such point, we get two possibilities: consider some magnitude that has sustained several divisions. Assuming that the aggregate size of the parts subject to division remains the same with that of the whole prior to division, we should now have more magnitude than we had to begin with: the magnitude of the parts undergoing division plus that of the points that survive division. But it does not seem right that repeated divisions should be making the magnitude divided bigger, which pushes us to the second possibility. As the divisions multiply, the aggregate size of the parts undergoing division is becoming smaller relative to the size of the original magnitude prior to division to account for the magnitude of the points that survive division, and with which the parts were together before the division was set under way.65

Unlike the first, the second possibility may appear to be receiving empirical support from the observation that fragments often seem to escape during the division of bodies, which would make the aggregate size of the parts of the body undergoing division smaller than that of the body prior to division.

But then, maybe something like sawdust comes to be as body is being divided, and in that way some body leaves the magnitude.

Here is Aristotle’s response to the possibility that these fragments are points escaping:

Even then the same argument applies: it [sawdust] is somehow divisible (316a34-b2).66

Cute as this argument may be, it proves nothing. It begins by assuming that magnitudinous

64Ὁµοίως δὲ κἂν ᾖ ἐκ στιγµῶν, οὐκ ἔσται ποσόν. Ὁπότε γὰρ ἥπτοντο καὶ ἓν ἦν µέγεθος καὶ ἅµα ἦσαν, οὐδὲν ἐποίουν µεῖζον τὸ πᾶν. Διαιρεθέντος γὰρ εἰς δύο καὶ πλείω, οὐδὲν ἔλαττον οὐδὲ µεῖζον τὸ πᾶν τοῦ πρότερον, ὥστε κἂν πᾶσαι συντεθῶσιν, οὐδὲν ποιήσουσι µέγεθος. (316a29-34; translation by Joachim, slightly changed). 65 According to Sedley “every division realized means one less contact” (p. 70), which would account for the aggregate size of the thing divided becoming smaller after every division. Then, “by the imposition of a division at point A, point A is replaced by two points” (p. 70), which would account for the aggregate size being larger. But if so, every contact lost by every division realized is getting replaced by the extra point gained by the imposition of that division, which should give us the same amount of points we had before, and, therefore, the same size. 66 Ἀλλὰ µὴν καὶ εἴ τι διαιρουµένου οἷον ἔκπρισµα γίνεται τοῦ σώµατος, καὶ οὕτως ἐκ τοῦ µεγέθους σῶµά τι ἀπέρχεται, ὁ αὐτὸς λόγος, ἐκεῖνο γάρ πως διαιρετόν. (at line 326b2 where I adopt Sedley’s (2004) p. 70 emendation: πως instead of πῶς.

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points survive division, which gives us two kinds of magnitude: (a) one that is subject to division and (b) that of the points that cannot be subject to division, since they are said to survive it. But magnitude not subject to division simply contradicts the divisibility thesis and no argument is needed to show that. What these lines succeed in doing is implicitly to make the claim, not argued or mentioned elsewhere in the chapter, that point is division.67 For, all they prove is that when we divide a magnitude in, say, two, all we have are the two parts and a division. Pointing out the obvious that the two parts are equal in size to the original proves that division, not point, is without magnitude. If that should be reason to say, as it is said here, that point has no magnitude, it must be because point is nothing more than a division.68

Aristotle makes the effort to present his own view because he is keen to dissociate it from the alternative on offer. And he confirms beyond doubt that the alternative conceives of point as a constituent of magnitude. For he goes on to ensure we understand that in his ontology point makes no contribution whatever to the production of magnitude; it presupposes it instead.

And if what is left was not body but a separate form or quality and the magnitude is points or contacts formed or qualified by them, it is absurd that a magnitude should consist of not magnitudes. Moreover, where will the points be and are they motionless or moving? And every contact is of two things, so that there is something besides the contact or the division or the point. (316b2-8). 69

Suppose that what escapes during division is not body or magnitude, but a form (εἶδος) or

67In the Metaphysics, Aristotle says explicitly that point is division (τοµή καὶ διαίρεσις) of line, line of plane and plane of solid (1060b14-17). It might seem peculiar that Aristotle equates the division of solids with point and not plane. Point, for him, is indivisible in all directions, whereas line and plane are indivisible in two and one respectively—point is magnitudeless, line has magnitude in one direction but lacks it in the other two and plane has it in two and lacks it in the third. Talking about points as opposed to lines or planes allows Aristotle to bring out his own conception of point, which further confirms that this is his real concern here. Doing so is, moreover, unproblematic. Point, line and plane all divide along the direction in which they are magnitudeless and, hence, themselves indivisible, which is what Aristotle wants to drive home. Being without magnitude is what makes them dividers. It is also unproblematic to speak of a point as being the division of a two or a three dimensional magnitude, given that infinitely many lines pass through any given point and infinitely many planes through any given line. 68 This answers a possible worry about the divisibility thesis. The Atomists can explain that division does not alter the size of the body it divides by pointing out that it is an exercise of separating bodily atoms from other such atoms with which they are conjoined. Division, for them, is not an intrusion on magnitude. For Aristotle, it is, which might worry some that it has a detrimental effect on magnitude. But the worry is unfounded, for the intrusion amounts merely to introducing non-magnitude into magnitude. This non-magnitude is not void, nor in anyway visible. So Aristotle aptly says: “ the point and all divisions, as well as anything which is indivisible in this way, declares itself to the mind as lack”. (ἡ δὲ στιγµὴ καὶ πᾶσα διαίρεσις, καὶ τὸ οὕτως ἀδιαίρετον, δηλοῦται ὥσπερ ἡ στέρησις. DA 430b20-1) 69 Εἰ δὲ µὴ σῶµα ἀλλ’ εἶδός τι χωριστὸν ἢ πάθος ὃ ἀπῆλθεν, καὶ ἔστι τὸ µέγεθος στιγµαὶ ἢ ἁφαὶ τοδὶ παθοῦσαι, ἄτοπον ἐκ µὴ µεγεθῶν µέγεθος εἶναι. Ἔτι δὲ ποῦ ἔσονται, καὶ ἀκίνητοι ἢ κινούµεναι αἱ στιγµαί; ἁφή τε ἀεὶ µία δυοῖν τινων, ὡς ὄντος τινὸς παρὰ τὴν ἁφὴν καὶ τὴν διαίρεσιν καὶ τὴν στιγµήν. (Translation by Joachim)

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quality (πάθος)—bodyness or magnitudeness, one might say. Forms or qualities are not magnitudes, so the question of further dividing the escapee cannot be raised. We also seem to avoid the threat of magnitude being extinguished; for though points do not have magnitude, magnitudeness or bodyness is still preserved as form or quality and can produce magnitude or body by informing or qualifying points.

Interestingly this argument makes use of non-Aristotelian doctrine. In, whatever way we may understand the term I render as ‘separate’ (χωριστὸν), forms and qualities are not separate according to Aristotle. Possibly he allows for the sake of argument that they might be to help us identify the friends of the conception of point he rejects. Still, even granting that forms and qualities are separate, point as Aristotle conceives of it cannot be a constituent of magnitude. Also for those who take them to be separate, forms form and qualities qualify magnitude, whereas point is magnitudeless, which brings us back full circle to the original absurdity that magnitude is comprised of non-magnitudes. Magnitude on this suggestion is allegedly generable out of something that cannot exist without it. Point is a division and the contact between the parts into which it divides, when it does. It cannot exist alone by itself apart from magnitude70 and is therefore unsuitable as a recipient of forms or qualities and cannot contribute to the production of magnitude in any way at all.

That points should be constituents of magnitude in the Aristotelian ontology is too eccentric a thought to deserve mention and refutation. But it is understandable that it should receive both, if the aim is clearly to dissociate this view from others in circulation that claim point is a constitutive of magnitude. And this is an aim Aristotle understandably should want to serve, given that these deeply entrenched views make nonsense of the divisibility thesis. In these lines Aristotle articulates, or, perhaps better, summarizes, a comprehensive conception: point is something, it is without magnitude, it is the division as well as the contact between the parts produced when it divides and consequently incapable of existing apart from magnitude. This will now be key to the central problem, which is coherently to characterize divisibility as a capacity of magnitude. 71

70 So, point is not χωριστὸν but always in something else (πάντα δὲ ταῦτα [sc. στιγµή, γραµµή, επιφάνεια] ἐν ἄλλοις ὑπάρχει καὶ χωριστὸν οὐδέν ἐστιν Metaph. 1060b15-7). 71 Having presented the reductio, Aristotle gives at 316b32 what would seem to be its conclusion, namely that magnitude must be composed of indivisibles, and further down: “such is the argument believed to establish the necessity of atomic magnitudes” (316b34-317a1). None of this implies that the reductio is an argument for atomism by Democritus, and nowhere do we see Aristotle, or anyone else, attribute such an argument to him. In fact a lengthy presentation of the motives for atomism in GC I.8 mentions only three: (a) movement and change,

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VI Before addressing the problem Aristotle states it in full in the aporia to whose resolution GC I.2 is dedicated:

We must therefore formulate the aporia anew. That every perceptible body is divisible at each and every point while at the same time also undivided is in no way absurd. For divisible it is in capacity, though actually it will remain undivided. But then, being simultaneously divisible everywhere in capacity would seem to be impossible. For if it were possible, it might actually occur, so that it is not simultaneously both actually undivided and divisible, but simultaneously divided at each and every point. Consequently, nothing will remain and the body will have passed-away into something bodiless, and would come-to-be again either out of points or absolutely out of nothing. And how is that possible?72

We are told that the aporia is formulated anew (ἐξ ἀρχῆς) so as to recall the preamble where it was previously signposted. Hence also the recurrence of the terms ‘δυνατόν’, ‘γένοιτο’, ‘ἅµα’ and ‘πάντῃ διαιρετὸν’ that were used then (316a16; 18). Here it is formulated as a full, two- horned paradox, the second horn of which contains the conclusion of the reductio. It goes a step further than the preamble in that the second horn is now explicitly said to be blocking the position asserted in Phys. III.6, where we were only told that the infinite exist but not actually. The same position is expressed here in the first horn with the statement that magnitude is undivided in actuality (ἐντελεχείᾳ). Magnitude is undivided in actuality because it can sustain an infinite number of divisions: no matter how many times it may have been divided already,

(b) the multiplicity of things that are, (c) the one-ness of Being (325a29-36). And, we saw Aristotle earlier attribute indivisibility of body to Leucippus or Democritus as opposed to indivisibility of planes he attributes to Plato, whereas the argument here allegedly proves indivisible magnitudes in general. It has moreover escaped attention that the lines leading up to this sentence offer two distinct considerations in favor of the existence of atomic magnitudes: “Οὔτε δὴ κατὰ µέρος διαιροῦντι εἴη ἂν ἄπειρος ἡ θρύψις, οὔτε ἅµα οἷόν τε διαιρεθῆναι κατὰ πᾶν σηµεῖον (οὐ γὰρ δυνατόν), ἀλλὰ µέχρι του.” (316b29-32). What we have here is an οὔτε … οὔτε construction setting apart distinct considerations “(1) neither could division into parts result in a process of ad infinitum breaking up, nor (2) can there be simultaneous division at every point (for that is not possible), but only up to a limit”. The first allows for the possibility of ad infinitum division, but finds it objectionable for the reason we detected previously, which Aristotle attributes to Leucippus, namely that the infinite breaking up violates the oneness of Being. It is the second consideration that refers to the argument of the reductio. They way these two are set apart, suggests that they are thought to be distinct, as they indeed are. The second does not allow division ad infinitum, while the first, and the only one Aristotle attributes to the Atomists in the GC, does. 72 Διὸ πάλιν ἐξ ἀρχῆς τὴν ἀπορίαν λεκτέον. Τὸ µὲν οὖν ἅπαν σῶµα αἰσθητὸν εἶναι διαιρετὸν καθ’ ὁτιοῦν σηµεῖον καὶ ἀδιαίρετον οὐδὲν ἄτοπον· τὸ µὲν γὰρ δυνάµει διαιρετόν, τὸ δ’ ἐντελεχείᾳ ὑπάρξει. Τὸ δ’ εἶναι ἅµα πάντῃ διαιρετὸν δυνάµει ἀδύνατον δόξειεν ἂν εἶναι. Εἰ γὰρ δυνατόν, κἂν γένοιτο, οὐχ ὥστε εἶναι ἅµα ἄµφω ἐντελεχείᾳ ἀδιαίρετον καὶ διῃρηµένον, ἀλλὰ διῃρηµένον καθ’ ὁτιοῦν σηµεῖον. Οὐδὲν ἄρα ἔσται λοιπόν, καὶ ἀσώµατον ἐφθαρµένον τὸ σῶµα, καὶ γίνοιτο δ’ ἂν πάλιν ἤτοι ἐκ στιγµῶν ἢ ὅλως ἐξ οὐδενός. Καὶ τοῦτο πῶς δυνατόν; (316b18-27).

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it can always be divided once more.

Commentators take the simultaneity requirement, which is explicitly made in the second horn, to be the mistake that generates the reductio. They take the solution to be to drop it and are frustrated that this is not how Aristotle solves it.73 In fact, the aporia itself is proof that this cannot be the solution. According to the first horn magnitude cannot be exhaustively divided, while the second horn insists that it needs to be. Surely, the resolution cannot be to dismiss the problematic horn simply because it conflicts with the favored one. There must be reason to insist on simultaneity, or else it would not be allowed to block what is after all said to be an intuitively plausible position.

This is stated in the first horn, and is as we previously diagnosed it. Magnitude can sustain an infinite number of divisions because it is divisible in capacity. But if so, the reductio argues, it must be possible for it to be exhaustively divided. Aristotle `is not prepared, and does well not, to abandon the capacity claim. He must therefore account for it in a way that does not make it vulnerable to the second horn, and begins by identifying a mistake committed by the reductio, which he says has so far escaped notice (λανθάνει). With that as a basis and the conception of point he has just laid out, he moves on to offer a proper characterization of divisibility. He does so in a complex and poorly understood passage.

(a) For since point is not contiguous with point, divisible everywhere is true of magnitude in one sense but not in another. (b) When this (i.e. that point is contiguous with point) is posited, it seems that there is a point anywhere but also everywhere, such that magnitude necessarily is divided into nothing; since there is a point everywhere, magnitude will consist of contacts or points. (c) The sense in which everywhere is true is that there is one point anywhere and all are like each one; but there are not more

73 So, according to Williams, (1982), 78, Aristotle fails to propose this solution and therefore does not solve the reductio. Another strategy is to put this solution in the text. Joachim translates “ὅτι µία ὁπῃοῦν ἐστι, καὶ πᾶσαι ὡς ἑκάστη” as “in so far as there is one point anywhere within it and all its points are everywhere within it if you take them singly one by one”, which is simply not in the Greek. According to Sedley (2004), pp. 80, too the solution is to prove that magnitude is “only potentially divisible to infinity”, i.e. precisely what Aristotle states in the first horn of the aporia only to block it with the second horn. Actually, Aristotle proved already in Phys. III.6, that an exhaustive division of magnitude is impossible by specifying a procedure that on the basis of which one can forever be cutting a finite magnitude into smaller parts. If you take, he says, a part of a magnitude which you get by dividing it on the basis of a given ratio, say 1/3, and keep adding parts to it by dividing the other part of the magnitude on the basis of that same ratio, you will never traverse the entire magnitude (206b7-9). There will always be some part left to divide. For any n number of such parts you take, there will still be one more such part to take. Of course, we may note, Democritus knows that. If he hadn’t thought of it himself, he would have learned from Zeno. It would be foolish of him to make himself vulnerable by producing an argument that denies it. The challenge Aristotle faces here is to ground the claim that magnitude can be subject to an infinite number of divisions by coherently characterizing divisibility as its capacity.

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points than one; for points are not in sequence, for if it [magnitude] were then divisible in the middle, it would be divisible also at the point contiguous to that; but marking is not contiguous with marking or point with point. (d) This, then, is division or composition, so that there is both dissociation and association, but not into atoms and from atoms (because the impossible consequences are many), nor in such a way that there can be a division everywhere (that would be possible only if point were contiguous with point), but there is dissociation into small and even smaller parts and association from such smaller parts.74

It is not the expression ‘divisible everywhere’ in itself that is problematic, which is also why Aristotle uses it throughout the GC. But there is a correct and a mistaken sense in which to use it, the implication being that the reductio uses it in the mistaken sense. In (a), Aristotle indicates the mistaken sense when he draws attention to the fact that points are not contiguous. That they are not contiguous he repeats twice in (c), thereby putting it beyond doubt that the mistake having so far escaped notice is that the reductio takes them to be contiguous. Why is this a mistake? Two things are contiguous, if their extremities are at the same place.75 Extremities are of parts of the things whose extremities they are, and so anything with extremities has parts. Parts are of magnitude. So, only things that have magnitude can be contiguous, and if points are assumed to be contiguous, they are assumed to have magnitude.

This mistake make it the case, we are told in (b), that the reductio takes points to be anywhere and everywhere (καὶ ὁπῃοῦν καὶ πάντῃ), such that magnitude is full of points everywhere (πάντῃ γὰρ εἶναι στιγµήν) and is made out of points (ὥστε ἐκ στιγµῶν εἶναι). In the mistaken

74 Ἐπεὶ γὰρ οὐκ ἔστι στιγµὴ στιγµῆς ἐχοµένη, τὸ πάντῃ εἶναι διαιρετὸν ἔστι µὲν ὡς ὑπάρχει τοῖς µεγέθεσιν, ἔστι δ’ ὡς οὔ. Δοκεῖ δ’, ὅταν τοῦτο τεθῇ, καὶ ὁπῃοῦν καὶ πάντῃ στιγµὴν εἶναι, ὥστ’ ἀναγκαῖον εἶναι διαιρεθῆναι τὸ µέγεθος εἰς µηδέν· πάντῃ γὰρ εἶναι στιγµήν, ὥστε ἢ ἐξ ἁφῶν ἢ ἐκ στιγµῶν εἶναι. Τὸ δ’ ἐστὶν ὡς ὑπάρχει πάντῃ, ὅτι µία ὁπῃοῦν ἐστι, καὶ πᾶσαι ὡς ἑκάστη· πλείους δὲ µιᾶς οὐκ εἰσίν· ἐφεξῆς γὰρ οὐκ εἰσίν, ὥστ’ οὐ πάντῃ· εἰ γὰρ κατὰ µέσον διαιρετόν, καὶ κατ' ἐχοµένην στιγµὴν ἔσται διαιρετόν· οὐ γάρ ἐστιν ἐχόµενον σηµεῖον σηµείου ἢ στιγµὴ στιγµῆς. Τοῦτο δ' ἐστὶ διαίρεσις ἢ σύνθεσις, ὥστ’ ἔστι καὶ διάκρισις καὶ σύγκρισις, ἀλλ' οὔτ' εἰς ἄτοµα καὶ ἐξ ἀτόµων (πολλὰ γὰρ τὰ ἀδύνατα) οὔτε οὕτως ὥστε πάντῃ διαίρεσιν γενέσθαι (εἰ γὰρ ἦν ἐχοµένη στιγµὴ στιγµῆς, τοῦτ' ἂν ἦν), ἀλλ' εἰς µικρὰ καὶ ἐλάττω ἐστί, καὶ σύγκρισις ἐξ ἐλαττόνων. (317a2-17; I put comma instead of period after “σύνθεσις” at 317b12 to indicate that what follows belongs to the consideration that starts in the sentence beginning with “Τοῦτο” in the same line. In lines 11-12 I translate ‘σηµεῖον’ as ‘marking’ to distinguish it from ‘στιγµὴ’, which I translate as ‘point’. I agree with Sedley (2004, 78, n. 27) that ‘σηµεῖον’ and ‘στιγµὴ’ refer to the same thing but that need not make them synonymous. I understand Aristotle here to be making the point that whether you call it a marking or a point you speak of the same thing: a division. Whether you call it, σηµεῖον or ‘στιγµὴ, σηµεῖον is not contiguous with σηµεῖον nor στιγµὴ with στιγµὴ. By using two terms to refer to the same thing, he puts emphasis on the claim that the thing referred to by either is not such as to be contiguous with another such thing. 75 This is the definition of ‘in contact’ (ἁπτόµενα) in Phys. VI.1 (231a22-3); ‘ἐχόµενα’ (contiguous) is used instead of ‘ἁπτόµενα’ in Phys. V.3, and is said to name a species of being in succession, namely being in succession (something to something else) and in contact (with each other). (ἐχόµενον δὲ ὃ ἄν ἐφεξῆς ὂν ἅπτηται 227a6). The definition of ‘in contact’ in Phys. VI.1 possibly makes that of ‘contiguous’ Phys. V.3 more precise.

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sense of ‘divisible everywhere’, magnitude is conceived of as being made out of points.

It is easy for this assumption to sneak in the argument unnoticed—hence the ‘λανθάνει’— if, as the case may be, it is based on a widely shared position on point.76 At any rate, we can see this assumption being guilty of a series of suspect moves that lead to the reductio. If magnitude is made out of points, point has magnitude, and a finite magnitude is made out of a finite number of points. This invites the idea that the task of performing all the divisions to which a magnitude is subject is completable. It is then but a small step to think of the requirement that capacities are fully actualizable in the subjects whose capacities they are as being met by completing a finite number of divisions.

Recounting these mistakes one by one is not to say that the reductio is making them one by one. The fundamental mistake, of which the others are a consequence, is to assume at the outset an overall conception of magnitude and point that contradicts the divisibility thesis. Identifying that conception in the very first move toward the solution of the reductio is Aristotle’s way of warning the reader not to confuse it with his. We need to make sure, then, that we read the solution in the light of this identification, which unfortunately still escapes attention, despite Aristotle’s warnings. Magnitude is not made out of points.77 And whatever makes this true makes it also true that magnitude does not contain points. It cannot be said to contain points, in Aristotle’s conception, for points are without magnitude and it is unclear what it would be for magnitude to contain non-magnitudes. Points are divisions and when they were in magnitude, they make it the case, not that it is composed of points, but that it is composed of the parts into which those points have divided it.78

76This is not the only place where Aristotle indicates that he has had to deal with objections against divisibility based on a conception of point as having magnitude. Cf. his complaint in Phys. VIII.8 (263a23-30) that treating the point that divides a line in two halves as being not one but two is to treat the line as being intermittent and not continuous. It is clear what he means. Treating the dividing point as two points implies that there are two points connecting the two halves into which a line is divided. They would, therefore, have to be contiguous. But for any two things to be contiguous, they would have to be magnitudinous. As in GC I.2, Aristotle’s point in Phys. VIII.8 is that continuity of magnitude is illegitimately treated in some quarters on the basis of assumptions that contradict it. 77 ἀδύνατον ἐξ ἀδιαιρέτων εἶναί τι συνεχές (Phys, VI.1, 231a24), though points are the termini of magnitude. 78But according to Sedley (2004), “Aristotle does not, as might at first appear, try to impose any qualification on the admission that there are points everywhere” (p. 78), which plainly ignores that Aristotle’s view of point does not allow that there be points in magnitude. And Joachim’s rendering of “ὅτι µία ὁπῃοῦν ἐστι, καὶ πᾶσαι ὡς ἑκάστη” as “in so far as there is one point anywhere within it and all its points are everywhere within it if you take them singly one by one”, implies that magnitude contains discrete points, which is nowhere in the Greek. Sedley goes so far as to emend πλείους δὲ µιᾶς οὐκ εἰσίν (ἐφεξῆς γὰρ οὐκ εἰσίν), ὥστ’ οὐ πάντῃ at 317a9 to πλείους δὲ µιᾶς οὐκ εἰσίν ἐφεξῆς (<ἐφεξῆς> γὰρ οὐκ εἰσίν), ὥστ’ οὐ πάντῃ, and translates: “but there are not more than one in sequence (for points are not ), and hence it [point] does not belong everywhere”. It is unclear what “and hence it [point] does not belong everywhere” means. More interesting is Sedley’s reason

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It is in (c) Aristotle introduces the correct sense of ‘divisible everywhere’: ‘there is one point anywhere’ (µία ὁπῃοῦν ἐστι) in magnitude. We are not to understand this, as it is often done, to mean that magnitude contains a point somewhere. If it contains one, it contains another, and if it contains two they could be contiguous. Such reading perpetuates the mistake about which Aristotle has already fired a warning, for it takes it that magnitude is full of points and we only need to pick anyone. Aristotle in fact alerts the reader not to slip into thinking such thoughts, when he adds the remark “not more than one” (πλείους δὲ µιᾶς οὐκ εἰσίν). When Aristotle says ‘there is one point anywhere’, I take him to mean that a point can be marked anywhere. Point is division, so that is his way of saying that magnitude can sustain a division just anywhere. It would not be true that magnitude could be divided just anywhere, if it contained points, even if it were full of them. For the positions at which it could be divided would then be determined by positions of the contiguous points. It could be divided only at the place where the points were or at their contact. To press home the idea that that magnitude can be divided just anywhere, Aristotle remarks about points in (c) that “all are like each one”. Any point marked anywhere in magnitude divides it as well as would any other, marked anywhere else in it. There are not more points than one because the one that is, when one in fact is, has divided it to give us two magnitudes—or makes the one that was be composed of two.

This sense of ‘divisible everywhere’, understood as can be divided just anywhere, sharply demarcates the divisibility thesis from that of the atomists. According to the latter, a magnitude composed of atomic particles, be those of sensible, intelligible body, can only be divided at the place of contact of these particles, whereas the particles themselves cannot be divided. According to the former magnitude is not composed of anything more basic; it itself is basic and can be divided anywhere. There is nothing in it that cannot be divided, for all is just magnitude. This sense of divisible everywhere also coherently characterizes divisibility

for the emendation: “obviously there are more points than one” in magnitude (n.25). But this is not “obvious” at all, nor that Aristotle would find acceptable the statement “there are not more [points] than one in sequence”. To be in sequence, a point would require one other point with which to be in sequence. This contradicts even Sedley’s emended bracketed statement, namely “for points are not ”. Most importantly, if points could be in sequence, nothing could preclude that they could also be contiguous, which contradicts Aristotle’s claim that points cannot be contiguous. The text of the manuscripts “there are no more [points] than one, for they are not in succession” makes perfect sense. It is already asserted that points cannot be contiguous. In itself, this does leave open the possibility that they could be in magnitude, not contiguously, but in succession. Still, having points in succession leaves open the possibility that they could be contiguous. So they had better not been in succession either.

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as a capacity of magnitude, in a way that observes the commitment on capacities as belonging to subjects: “it is possible for it [‘it’ = ‘the subject’s coming to be in actuality that which it is said to be in capacity’] to obtain without qualification at some point in time”.79 Having divisibility actualized in magnitude is simply for magnitude to be divided just anywhere. This is a completable task, requiring nothing more than marking a point anywhere in that magnitude.

The aporia is resolved. First, the statement of the second horn is refuted, not just dismissed. That horn had raised an obstacle premised on contradictory premises, fetched from conflicting theses. One of these theses supplies magnitude made out of points. The other supplies points without magnitude and conceived of as being incapable of being constituents of magnitude.80 So, the intuitive plausibility of the other statement of the first horn, namely that magnitude can be divided so many times that it will never be divided all these times, suffers no damage. Better still, the two propositions of the first horn are shown to be complementary and in the right way. Magnitude is never divided in actuality. Whenever divisibility is actualized in some magnitude it produces parts, which are also divisible in capacity, as is also the case with the parts produced by bringing to actuality the capacity of those parts.

In fact, bringing to actuality the capacity specified above in any finite magnitude m, and subsequently in the parts thus produced, and continuing to do so in all subsequent parts of parts, makes the number of parts that are thus produced larger than infinite. For everything that is true of m is also true of each and every one of its parts. Hence, the cardinal number of the set of parts that dividing any m can produce is larger than that of the set of natural numbers. Unbeknownst to him, probably, Aristotle has discerned a non-denumerable set.

We may now turn to (d). The result of marking a point in a unit of magnitude may be characterized as ‘division’ or ‘composition’: the unit is divided in two (or any given number of) smaller parts, but having thus been divided it is also composed of these parts.81 This, I suggest, is the sense of the strange sounding sentence “now this is division or composition”

79 ἐνδέχεται ἁπλῶς ἀληθεύεσθαί ποτε. Met. 1048b12-3 80 Mathematicians nowadays do conceive of the continuum as a set of densely ordered unexpended elements, i.e. what Aristotle calls points, but this is not an idea Aristotle shares. 81 Aristotle uses the terms division (διαίρεσις) and composition (σύνθεσις) to make a similar point also in other contexts. See DA 430a30-b4 where it is said that a proposition is both a composition and a division. It is a composition of the elements that go together to constitute it and the same proposition is also divided into those elements.

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(317a12) that introduces (d).82 It is important for Aristotle to point this out here because he can thereby account for processes that share a characteristic of association (σύγκρισις) and dissociation (διάκρισις), as he attributes these processes to the Atomists. According to this attribution, we recall, atoms join together in association and come apart during dissociation. The feature of these operations, of interest to Aristotle here, is that each of the elements involved retains its identity and distinctness during both association and dissociation. For the Atomists this is a consequence of the unalterability of the atoms. Aristotle wants to allow for associations and dissociations, i.e. operations in which the elements involved retain their identity and distinctness, while also denying that there exist indivisible magnitudes. This is what he does when he points out that a unit of magnitude may be divided into smaller parts that then compose it. The point made in (d) can be illustrated with one of Aristotle’s favorite examples. Consider a wall constructed by associating pieces produced by dividing different magnitudes (pieces of wood and stone, say), as indicated in (d). In accordance with (d), we can say that this wall is composed of, or divided into, wood and stone. It can also be deconstructed into these parts, if the pieces of wood and the stones, the association of which had generated it, are taken apart one by one. So we can have association or dissociation without the constituent elements involved having to be indivisible; they need only be small or smaller pieces of divisible magnitude.

The aim of the discussion on divisibility in GC I.2 is not to refute a Democritean argument for atoms. That is already well refuted. Moreover, the reductio could not have been an argument

82 “τοῦτο δ' ἐστὶ διαίρεσις ἢ σύνθεσις”. I take “τοῦτο” to be taking us back to section (c), where Aristotle solves the reductio. Based on the lesson from (c), it is clear that introducing a point in a unit of magnitude divides it in two smaller units, and the original unit is now composed of these two. More points would divide it in more smaller units, and it would then be said to be composed of more units. This thing is both division and composition. So I take διαίρεσις to be naming what we get when we divide a unit of magnitude into smaller units and ‘σύνθεσις’ to be these units considered together as forming the whole original unit. This is normal Aristotelian usage. ‘Division’ (διαίρεσις) and ‘composition’ (σύνθεσις) are used to make exactly the same point in DA 430a30-b4. The sentence “τοῦτο δ' ἐστὶ διαίρεσις ἢ σύνθεσις” has puzzled commentators. Joachim (1922) thinks it is possibly displaced (p. 86), whereas Sedley (2004) takes σύνθεσις to be another name for point; He translates: “it is a point that serves as a division or joint” (p78). However, such a remark at this juncture is entirely out of place and leaves hanging the “Ὥστε” which begins the next sentence. More importantly, nowhere does Aristotle use ‘σύνθεσις’ to refer to point. Other than διαίρεσις (division), Aristotle says that a point is also ἀφή (contact), never ‘σύνθεσις’. On the other hand he systematically uses ‘σύνθεσις’ in the GC to characterize wholes composed of constituent parts that retain their identity and distinctness as the parts they are, within the whole they compose. So, at 315a23, and more strikingly at 334a27, ‘σύνθεσις’ names wholes formed of parts arranged in the way stones and bricks are arranged in a wall. At 328a6, b12 and 19 ‘σύνθεσις’ is contrasted to µίξις, the characteristic of which, we recall, is that its constituent parts fuse in a homogeneous entity where the parts are never preserved actually. More importantly, elsewhere in the GC, for instance at 327a18 and 328a15, where ‘διαίρεσις’ occurs together with ‘σύνθεσις’, as they do here, ‘διαίρεσις’ indicates an operation in which constituent elements comprising such wholes as referred to by ‘σύνθεσις’ can be removed one by one, as brick and stone can be removed from a wall.

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by Democritus, and no independent evidence suggests that he ever offered anything like it. The reductio in GC I.2 is not aimed at refuting anything. It only aims at propagating a conception of magnitude and point that must have seemed as alien then as it does now. The idea of considering and studying the sensible world, independently of what makes it a subject- matter for physics, i.e. exclusively as spatial extension and claiming that in so doing one studies a fundamental feature of the world and not merely a fictitious object will surely have seemed strange. Even more strange must have seemed Aristotle’s non-foundationalist conception of the world thus considered, i.e. the idea that it is not composed of more basic components, be those points or whatever else. This seems a strange idea also now, which may be why interpreters expect Aristotle to be thinking of magnitude as somehow being composed of and containing points. It does not, for him. Point for Aristotle is only brought into existence by the act of division, whereas both among his contemporaries and now point is thought to be a constituent of magnitude; then as something that has magnitude, now as something unextended on which continuous magnitude supervenes. The key notion Aristotle aims at passing on to the reader in GC I.2 is that it is a mistake to think of divisibility in terms of points. The right way to think of it is in terms of parts. Magnitude is divisible not because it contains a zillion points but because it can be split up into parts without end. For, “nothing among what is continuous is divided into indivisibles” (οὐθὲν ἦν τῶν συνεχῶν εἰς ἀµερῆ διαιρετόν 231b11-12) be those indivisibles with or without magnitude. Anything that is divisible is divided always in divisibles. That is what it is to be continuous (εἰ διαιρετόν, εἰς ἀεὶ διαιρετά· τοῦτο δὲ συνεχές 231b14-15)

Appendix At 316b8-14 begins a sentence that appears to be summing up the reductio.

“These, then, are the difficulties resulting from the supposition that any and every body, whatever its size, is divisible everywhere.” (Εἰ δή τις θήσεται ὁτιοῦν ἢ ὁπηλικονοῦν σῶµα εἶναι πάντῃ διαιρετόν, πάντα ταῦτα συµβαίνει. 316b8-9).

Here is what follows this summation:

One may look at it also this way. If, having divided a piece of wood or anything else, I put it together, it is again equal to what it was, and is one. Clearly this is so, whatever the point at which I cut the wood. Therefore, it has been divided potentially everywhere. What, then, is there besides the division?

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And even if we suppose there is some quality, yet how is it dissolved into such constituents and how does it come-to-be out of them? Or how are such constituents separated so as to exist apart from one another?” (Ἔτι ἐὰν διελὼν συνθῶ τὸ ξύλον ἤ τι ἄλλο, πάλιν ἴσον τε καὶ ἕν. Οὐκοῦν οὕτως ἔχει δηλονότι κἂν τέµω τὸ ξύλον καθ’ ὁτιοῦν σηµεῖον. Πάντῃ ἄρα διῄρηται δυνάµει. Τί οὖν ἔστι παρὰ τὴν διαίρεσιν; εἰ γὰρ καὶ ἔστι τι πάθος, ἀλλὰ πῶς εἰς ταῦτα διαλύεται καὶ γίνεται ἐκ τούτων; ἢ πῶς χωρίζεται ταῦτα;.

These lines (316b9-14) have received special attention following Prantl’s claim that they should be excised. The argument for excising them is that the preceding sentence prepares the reader for a statement of the conclusion of the reductio instead. A concluding statement does not appear before later. So, Prantl claims, 316b9-14 interrupts the flow of the argument, and must be by a later hand.83 Sedley too claims that these lines are out of place, but agrees with Joachim that they are genuinely Aristotelian. So, he transposes them.84 According to Sedley, the reductio up to 316b16, but excluding lines 316b9-14, reports an historically correct version of an argument by Democritus, whereas lines 316b9-14 offer an anachronistic version of an argument for atomism written by Aristotle or an Aristotelian in Aristotelian terminology, and should be placed between lines 316b27 and 316b28.

There is no support in the tradition for excising these lines. And it is less than clear for what purpose a later hand should add them. They offer no new argument. Nor is there any reason for transposing them. Moreover, Sedley’s suggestion is problematic on several other counts. He claims not only that the reductio up to 316b16 (except lines 316b9-14) is an historically accurate reproduction of Democritus’ argument, which as we have seen is unlikely, but also that lines 316b14-19, which in Sedley’s reconstruction come before 316b9-14, are a report on Democritus’ own thinking. Here are these lines:

“Since, therefore, it is impossible for magnitudes to consist of contacts or points, there must be indivisible bodies and magnitudes. Yet, if we do postulate the latter, we are confronted with no less

83 Prantl (1857), 490, n 15. Also Williams (1991), 72; contra Joachim (1926), 82. 84Joachim (1926), 82 concedes to Prantl that it is difficult to connect these lines to what has gone before, but thinks that bracketing them is unnecessary. His solution is to say that they take the argument a step further because the clause πάντῃ ἄρα διήρηται δυνάµει (316b11-12) refers to a potential division everywhere, whereas previously it was assumed that the divisions everywhere were actually performed. Surely, though, δυνάµει here cannot but have the same sense it does throughout the argument, which is to imply the presence of a capacity that can be fully actualized, which in present context is understood to mean being exhaustively divided. In fact, the text positively confirms that it is so understood also here. For, it states explicitly that there comes a point where there is no more wood left but just divisions, and the only way in which we know this can be the case is if the wood actually sustains the divisions it can sustain δυνάµει.

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impossible consequences. Those we have examined in other works. But we must try to disentangle these perplexities, and must therefore formulate the whole problem over again.” (ὥστ’ εἴπερ ἀδύνατον ἐξ ἁφῶν ἢ στιγµῶν εἶναι τὰ µεγέθη, ἀνάγκη εἶναι σώµατα ἀδιαίρετα καὶ µεγέθη. Οὐ µὴν ἀλλὰ καὶ ταῦτα θεµένοις οὐχ ἧττον συµβαίνει ἀδύνατα. Ἔσκεπται δὲ περὶ αὐτῶν ἐν ἑτέροις. ἀλλὰ ταῦτα πειρατέον λύειν, διὸ πάλιν ἐξ ἀρχῆς τὴν ἀπορίαν λεκτέον (316b14-19), Joachim’s translation).

Allegedly we are here witnessing Democritus concluding his own argument with the remark that though it is necessary for there to exist indivisible bodies and magnitude in general, this view has impossible (ἀδύνατα) consequences. But there is no evidence to support that Democritus held anything else than body to be indivisible, or that he ever expressed such doubts about his own view. Moreover, lines 316b14-19 contain an aporia, the second horn of which claims that atomism is impossible. If these lines present Democritus’ own aporia, as Sedley maintains, we should expect an attempt by Democritus at removing the impossibilities. We get nothing of the sort. It can only be Aristotle who here says about himself that he has already shown Democritus’ view to have impossible consequences. He makes the exact same remark at 317a14, with unambiguous reference to the Atomists. What Aristotle does in this horn of the aporia is to remind the reader that, as he has demonstrated elsewhere, the atomist thesis has impossible consequences. He does it to make it clear that the alleged necessity mentioned in the first horn does not represent a threat, for in that way to motivate the upcoming refutation of the reductio. He actually implies that this is what he does in the last sentence of the passage above, by saying that he will clearly flash out the problem by formulating it anew, then doing so and subsequently solving it.

Even if we were to accept that what we have here is a broken Democritean promise to address the impossible consequences of atomism, we should at least be getting additional argument against divisibility, albeit in Aristotelian terms. According to Sedley, what we get is Aristotle allowing Democritus to “acknowledge an Aristotelian countermove to the atomist argument, with a view to rebutting that countermove in the next sentence.” Allegedly, the countermove acknowledged is to point out that to call a magnitude divisible everywhere is “no more than to indicate a potentiality”. The rebuttal is that “to call a state of affairs ‘potential’ is to allow that it could be actual.”85 Surely, though, if this is the alleged rebuttal, it does nothing more than restating what the preamble established (316a14-23) and the reductio has been assuming all along, namely that divisibility is a capacity of magnitude. In fact, the statement at 316b23-24,

85 Sedley (2004), 74

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that if something is possible, it might actually occur (εἰ γὰρ δυνατόν, κἄν γένοιτο), alleged to be part of the anachronistic Democritean rebuttal, makes the exact same point made in that preamble, and in identical terms. Since Sedley takes the argument of the lines he transposes (316b9-14) to be for a Democritean position, he accounts for its Aristotelian looks by saying that it is not part of the reductio and that it is offered on Democritus’ behalf by Aristotle or an Aristotelian. But whatever makes the argument in these lines look Aristotelian makes the rest of the reductio look exactly the same.

The lines 316b9-14 connect smoothly with what has gone before; ‘πάθος’ at 316b13 clearly points back to this same term at 316b3, and it is only by linking it to that previous occurrence we understand that ‘wood’ in these lines is to be thought of as a quality, in the way it was proposed previously that body could be a quality. And though Prantl is right to point out that they add no new argument, it is not true that they add nothing. Up to this point we have seen an abstract theoretical argument run on technical terms: magnitude, body and points. 316b9- 14 sums up all (πάντα ταῦτα) the alleged consequences (συµβαίνει) of the divisibility thesis, according to this argument, by presenting, for the reader to consider, a concrete example.86 Instead of magnitude or body we have a piece of wood; instead of points we have divisions.

The important aim served by these lines is to summarize while illustrating the results of the previous argument. They throw into stark relief Aristotle’s conception of magnitude and point, which is particularly important if, as the case seems to be, both are novel and alien. It also lends that conception intuitive support. All you have, if you break a piece of wood in two, are two smaller pieces of that wood. Anywhere you break it, this is what you get. And if you break it once you know can break many times, even though you do not actually do it, because you did it already with exactly this piece of wood. However many times you break it you’ll be getting exactly what you got first time, just more of it: more pieces of wood and more divisions; and points are just that: divisions. If you’re committed now, as you’ve agreed that you are, that the breaking comes to an end, that will have to be because there is no more wood left. For you know that wood you can break. All you’ll have at that stage then is divisions, and it is just not clear how you’ll be getting wood out of that.

86 Hence, the ἔτι introducing the lines 316b9-14 invites the reader to consider the results of the argument once more, this time concretely illustrated on a some piece of wood.

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